Strongly stable generalized finite element method

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ScienceDirect Comput. Methods Appl. Mech. Engrg. 327 (2017) 58–92 www.elsevier.com/locate/cma

Strongly stable generalized finite element method: Application to interface problems Ivo Babuška a , Uday Banerjee b , ∗, Kenan Kergrene c a ICES, University of Texas at Austin, Austin, TX, United States b Department of Mathematics, 215 Carnegie, Syracuse University, Syracuse, NY 13244, United States c Department of Mathematics and Industrial Engineering, Polytechnique Montréal, Canada

Available online 18 August 2017

Highlights • • • • •

Detailed mathematical analysis of the GFEM for 2D interface problems. Verifiable sufficient conditions to guarantee a well-conditioned GFEM. Proof of the well conditioning of GFEM when applied to 2D interface problems. The notion of Strongly Stable GFEM that ensures efficient iterative solvers. Proof of the optimal convergence of GFEM applied to 2D interface problems.

Abstract In this paper, we present the theoretical justification for the Stable Generalized Finite Element Method (SGFEM) when applied to smooth interface problems. We prove that a Generalized Finite Element Method (GFEM) is stable, i.e., its conditioning is (a) not worse than that of the Finite Element Method, and (b) robust with respect to the mesh, if the enrichment space of the GFEM satisfies two axioms. We provide element based sufficient conditions on the enrichment space that will guarantee that the axioms are satisfied. We show that the enrichment space of the GFEM, used to address the interface problems in 2D, satisfies the sufficient conditions and thus the two axioms yielding an SGFEM. The idea of a strongly stable GFEM associated with one of the two axioms has been introduced; strong stability of the GFEM is important for designing efficient iterative methods to solve the underlying linear system. A proof of the optimal convergence of the GFEM for the interface problem has also been derived in this paper. The work in this paper is the continuation of Kergrene et al. (2016), where the stability of the GFEM, when applied to interface problems, was established through numerical experiments. The numerical results in Kergrene et al. (2016) indicated that the GFEM is indeed strongly stable. c 2017 Elsevier B.V. All rights reserved. ⃝

Keywords: GFEM; XFEM; SGFEM; Interface; Conditioning; Robustness

∗ Corresponding author.

E-mail address: [email protected] (U. Banerjee). http://dx.doi.org/10.1016/j.cma.2017.08.008 c 2017 Elsevier B.V. All rights reserved. 0045-7825/⃝

I. Babuška et al. / Comput. Methods Appl. Mech. Engrg. 327 (2017) 58–92

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1. Introduction The Generalized Finite Element Method (GFEM) has been successfully applied over the last 20 years to a wide variety of stationary or time dependent engineering problems with non-smooth solutions, e.g., problems involving moving interfaces, crack propagation, material discontinuity, solid–fluid interactions, etc. We refer to the review articles [1–3] and to the references therein for various work on GFEM and its applications. The method has also been incorporated into commercial codes, e.g., Abaqus and LS-DYNA [4,5]. It is also known as the Extended Finite Element Method (XFEM) in the literature [6–8]. The GFEM is an instance of the more general Partition of Unity Method (PUM) where the standard finite element hat functions are used as the Partition of Unity (PU). The PUM was developed in mid-nineties [9–11] and its impact on the Computational Science and engineering computations is clearly visible, for example, in the total citations of these 3 papers. The PUM/GFEM allows to extend, locally, the space of trial functions by special functions mimicking the local property of the exact solution through a PU. A typical use of these methods will be on an interface problem, where the gradient of the solution has jump along the interface; additionally, the solution could also have singularities when the interface is non-smooth. The PUM or GFEM does not need special meshing and the convergence is not affected by the non-smooth features of the solution. We mention that the Method of Cloud [12–14] is very similar to GFEM. The trial space of GFEM is the direct sum of two spaces – the classical piecewise linear finite element space S F E M and an enrichment space S E N R . The support of the shape functions of both these spaces are the standard “finite element stars” subordinate to a simple mesh. Typically, the cardinality d E N R of the shape function of S E N R is much smaller that the cardinality d F E M of the shape function of S F E M . For example, for a smooth interface problem in 2-D, we have 1/2 d E N R = O(d F E M ). The enrichment space is constructed either using the available analytical information about the solution or numerically computed solutions of certain local problems [15]. The stiffness matrix of the GFEM could be naturally written in the block form [ ] A A12 A = 11 A21 A22 where A11 , A22 are the stiffness matrices associated with only S F E M , S E N R , respectively; the matrices A12 , A21 depend both on S F E M and S E N R . We note that A11 is the standard finite element stiffness matrix. One of the main difficulties in GFEM is that it could be badly conditioned, namely, its conditioning could be much worse than that of the standard Finite Element Method. In other words, the scaled condition number K(A) of the stiffness matrix A could be much larger than K(A11 ). One of the main reasons for this problem is related to the “angle” between the spaces S F E M and S E N R . This is an extremely important issue in GFEM as it does not only affect the rounding error but affects the efficiency of iterative methods to solve the underlying linear system when A is large, e.g., in elasticity or thermo-elasticity problems in 3-D. For various applications, suitable enrichment spaces S E N R have been constructed and reported in the literature and their effectiveness have been established through the analysis (mostly through computations) of convergence of the associated GFEM. The conditioning problem of GFEM has also been observed and addressed in [16–24]. We further mention that a method related to the GFEM, referred to as the Improved XFEM, has been reported in [25–27]. This method requires less degrees of freedom than the standard GFEM and the numerical experiments show that the method yields optimal convergence and is not badly conditioned when applied to problems with non-smooth solutions. However, systematic analysis of the stability and robustness of the methods employing enrichments is rare in the literature. To address these issues, the idea of a Stable GFEM (SGFEM) was developed in [28] and further studied in [29–31]. In this paper, we introduce a related notion of a Strongly Stable GFEM. A GFEM is an SGFEM if it yields optimal convergence and K(A)/K(A F E M ) ≤ L < ∞ with L independent of the mesh. An SGFEM is strongly stable if, in addition, K(A E N R ) ≤ K < ∞ with K independent of the mesh. It was shown through numerical experiments in [32] that the GFEM, applied to smooth interface problems, is indeed an SGFEM. The numerical experiments in [32] also indicated that the GFEM is strongly stable (though the qualifier “strongly stable” was not used). The notion of strong stability of GFEM is particularly important for efficiently solving the underlying linear system by an iterative method, which was clearly established in [32]. This paper is a continuation of [32]. Here we present the theoretical framework to explain the numerical results presented in [32]. We analytically prove that the GFEM, when applied to smooth interface problems in two-dimensions (2D), is indeed strongly stable, i.e., it is robust with respect to the position of the mesh relative to the interface. In

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particular, we show that a GFEM is strongly stable if the enrichment space S E N R satisfies two axioms; there axioms were reported in [28,31]. We have presented “element based” sufficient conditions such that the axioms are satisfied. These sufficient conditions are quite general and could be used in constructing S E N R , yielding a strongly stable GFEM, for other application problems. A suitable S E N R for the non-smooth interface problem, where the solution with discontinuous gradients may also have singularities, and the strong stability of the associated GFEM will be presented in a forthcoming paper. The paper has been organized as follows. In Section 2 we present the model problem and we describe the GFEM and set various notations in Section 3. The proof of the optimal convergence of GFEM is presented in Section 4. In Section 5, we present the precise notion of strong stability of GFEM through two axioms and “element based” sufficient conditions such that the axioms are satisfied. In Sections 6 and 7 we prove that the sufficient conditions hold for S E N R used for the smooth interface problems. 2. The interface problem Let Ω ⊂ R2 be a bounded, simply connected domain with a smooth boundary ∂Ω . Let Γ be a smooth interface that divides Ω into two domains Ω1 and Ω2 such that Ω = Ω 1 ∪ Ω 2 and Γ = Ω 1 ∩ Ω 2 . We are interested in the weak solution u of the problem: −∇ · (a ∇u) = f, a ∇u · n⃗ = g N ,

in Ω ,

(2.1)

on ∂Ω ,

(2.2)

where n⃗ is the outward normal to ∂Ω . The function a(x) is bounded, i.e., 0 < β0 ≤ a(x) ≤ β1 < ∞,

(2.3)

and is discontinuous along Γ given by a(x) = ai ∈ R,

for x ∈ Ωi , i = 1, 2.

It is clear that β0 ≤ ai ≤ β1 for ⏐ i = 1, 2. The results of this paper will be same if, instead of the piecewise constant function a(x), one considers a ⏐Ω ∈ C 2 (Ω i ). We assume that f is smooth in Ω , g N (s), s ∈ ∂Ω is smooth on ∂Ω ; in i particular, g N (s) ∈ C 1 (∂Ω ), where s is the arc length parameter. Also, a∇u · ν⃗ is continuous across the interface Γ , where ν⃗ is the normal to the interface Γ . The weak solution of (2.1)–(2.2) is characterized by u ∈ E(Ω ) := {u ∈ H 1 (Ω ) : B(u, u) < ∞} B(u, v) = F(v), for all v ∈ E(Ω )

(2.4)

where B(u, v) =

∫

∫ a ∇u · ∇v d x, Ω

F(v) =

f v dx +

Ω

∫ ∂Ω

g N v ds

and the data f , g N satisfy the compatibility condition F(1) = 0. We note that the solution u of the problem (2.4) is unique up to a constant. We will consider two types of interface Γ in this paper – a straight interface and a smooth closed interface. If Ω1 ⊂⊂ Ω is a simply connected domain and Ω2 = Ω \ Ω 1 , then Γ = Ω 1 ∩ Ω 2 = ∂Ω1 is the closed curved interface. For the straight interface, we have Ω 1 ̸⊂ Ω , Ω ⏐ 1 ∩ Ω2 = ∅ and Γ1 = Ω 1 ∩ Ω 2 . The solution u ∈ E(Ω ) of (2.4) satisfies u ⏐Ω ∈ H 2 (Ωi ) ∩ W∞ (Ω i ) for i = 1, 2 and could be written as i

u = u0 + χ H V

(2.5)

where – u 0 ∈ H 2 (Ω ). – χ is a smooth cut-off function with support Ω 0 , where Ω0 is a K -neighborhood of Γ such that ν⃗ (s1 ), ν⃗ (s2 ), s1 , s2 ∈ Γ do not intersect in Ω0 . Furthermore, χ(x) = 1,

x ∈ Ω0,0 where Γ ⊂ Ω0,0 ⊂ Ω0

(2.6)


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– H is a smooth function in Ω which is constant (̸= 0) along the normal ν⃗ (s), s ∈ Γ in Ω0 ; – V is the distance function given by V (x) := dist(x, Γ )

(2.7)

Note that V (x) = 0 for x ∈ Γ and V is smooth in Ωi ∩ Ω0 , i = 1, 2, i.e., V := max{∥V ∥W∞ 2 (Ω ∩Ω ) , ∥V ∥W 2 (Ω ∩Ω ) } < ∞. ∞ 0 0 1 2

(2.8)

The straight interface Γ intersects ∂Ω at two points and the solution u ∈ E(Ω ) of (2.4), in general, will have singularities at those points. In this paper, we assume that the data⏐ is such that the solution u ∈ E(Ω ) of (2.4) is 1 continuous in Ω , does not have singularities at the points Γ ∩ ∂Ω , u ⏐Ω ∈ H 2 (Ωi ) ∩ W∞ (Ω i ) for i = 1, 2, and could i be written as (2.5). We note that for the straight interface, the distance function V (x) = dist(x, Γ ) is linear on Ωi ∩ Ω0 , i = 1, 2, and V (x) = 0 for x ∈ Γ . We will address the situation where Γ is non-smooth and the solution u has singularities in a forthcoming paper. Remark 2.1. We note that in a more general interface problem, f in (2.1) could be replaced by f + qδ(Γ ), where δ(Γ ) is the Dirac function on Γ , q(s), s ∈ Γ is smooth on Γ . However, this only affects the right hand side of (2.4) and the compatibility condition, and does not affect the choice of the enrichment functions employed in the GFEM and the features of the method that we plan to study in this paper. We note that in this case, a ∂u (⃗ν is normal to Γ ) ∂ ν⃗ will not be continuous across Γ , but it does not affect results presented in this paper. □ In the rest of the paper, we will write E := E(Ω ). The energy norm of v ∈ E is given by ∥w∥E := B(w, w)1/2 . Moreover for any A ⊂ Ω , we set ∫ a∇v · ∇w dx. B A (v, w) := A

Based on this notation, we will use E(A) := {w ∈ H 1 (A) : ∥w∥E(A) < ∞} where ∥w∥E(A) := B A (w, w)1/2 . m We will also use the standard notations for the norms and semi-norms of the Sobolev spaces H m (A), W∞ (A), m = 1, 2, denoted by

∥v∥m,A := ∥v∥ H m (A) , |v|m,A := |v| H m (A) , m (A) , m (A) . |v|m,∞,A := |v|W∞ ∥v∥m,∞,A := ∥v∥W∞ We note that energy norm ∥ · ∥E(A) and the semi-norm |·|1,A are equivalent from (2.3). Furthermore, we will use |A| to denote the area of the set A. If I is a discrete set, we will use |I | to denote the cardinality of I . 3. The stable generalized finite element method Let Th be a standard finite element mesh on the domain Ω satisfying the standard minimal angle condition, where 0 < h < 1 is the discretization parameter. We set the following notation associated with the mesh Th : ◦ Let {xih : i ∈ Ih } be the set of finite element nodes of Th , where Ih is the index set of the nodes and |Ih | := card{Ih }. ◦ Let {τsh : s ∈ E h } be the set of finite elements associated with the mesh Th . The elements τsh are closed triangles and E h is the index set of the elements. The interior of τsh will be denoted by τ˚sh . ◦ For an element τsh , s ∈ E h , we let Ihs ⊂ Ih be the index set of the nodes (vertices) of τsh . Thus {xih : i ∈ Ihs } are the nodes of the element τsh .

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Fig. 1. The interface Γ is the dashed curve. The perturbation Γh in an element is given by the bold line. (a) Γ intersects two sides of an element once; (b) Γ intersects two vertices of an element; (c) Γ intersects only one side of one element and on three sides of an adjacent element; Γ intersects both sides of an element, but intersects one sides multiple times.

◦ For i ∈ Ih , we define E hi := {s ∈ E h : i ∈ Ihs }. Thus the set {τsh : s ∈ E hi } is the collection of all the elements with the common node xih . ◦ Let Nih be the standard piecewise linear finite element hat function associated with the node xih , i ∈ Ih with the support ⋃ ωih = τsh . (3.1) s∈E hi

The open sets ωih , i ∈ Ih , are called the patches. Note ωih is the closure of ωih . We set S1h := span{Nih , i ∈ Ih }.

(3.2)

The space S1h is the standard finite element space of piecewise linear functions associated with the mesh Th . The approximation space SG F E M of a Generalized Finite Element Method (GFEM) is obtained by augmenting the standard finite element space S1h with an enrichment space S2h . The definition of S2h • for the closed smooth curved interface Γ , depends on the perturbations of Γ associated with the mesh Th ; • for the straight interface Γ , does not depend on any perturbations of Γ . We remark that for the curved interface, it is not necessary to use a perturbation of Γ to define S2h . However we use it because it is computationally simpler. The enrichment space S2h also depends on a specific continuous piecewise linear function V˜h associated with the distance function V ; the function V was defined in (2.7). We will discuss these notions before defining S2h . We will first define the function V˜h and then S2h . ⏐ The function V˜h when Γ is curved: In this case, the function χ V is continuous in Ω and χ V ⏐Ω ∩Ω , i = 1, 2 is i 0 smooth but nonlinear. The function V˜h , which depends on χ V , is defined relative to a perturbation of Γ , which we discuss first. The perturbation of the interface Γ is a closed polygonal curve Γh defined as follows: For an element τsh such that τ˚sh ∩ Γ ̸= ∅ and Γ intersects two sides of τsh with each side intersected only once, then Γh |τsh is obtained by joining these two points by a straight-line; see Fig. 1a. If Γ intersects two vertices of the element, then Γh coincides with the edge of the element between the two vertices; see Fig. 1b. However if Γ intersects only one side of τsh , then Γh ∩ τsh = ∅ as we see for the element τ1 in Fig. 1c; Γh is defined in the adjacent element τ2 . In fact, Γ may intersect all the three sides of an element as τ2 in Fig. 1c; Γh is a straight line as shown in this figure. Moreover, if Γ intersects two sides of the element but has multiple intersections with one of the sides, one of the intersection points near the middle of that side is used to define Γh ; see Fig. 1d. Thus Γh is a piecewise linear interpolant of Γ ; it depends on the mesh parameter h. Also note that Γh is a closed curve.

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Fig. 2. (a) The interface Γ (dashes curve) divides the element τ into τ s, 1 and τs,2 . The perturbed interface Γh (thick line) divides the element τ T and τ Q = τ Q,1 ∪ τ Q,2 . into τs,1 s,2 s,2 s,2

Let Ω1,h ⊂⊂ Ω be the domain such that ∂Ω1,h = Γh and set Ω2,h = Ω \ Ω 1,h .

(3.3)

Clearly Ω1,h , Ω2,h depend on h, Ω1,h ⊂⊂ Ω , and they are perturbations of the domains Ω1 , Ω2 respectively. We define L h = (Ω 1,h ∩ Ω 2 ) ∪ (Ω 2,h ∩ Ω 1 ).

(3.4)

It is clear that L h is the union of “lens” shaped regions L h ∩ τsh along the curve Γ , where |L h ∩ τsh | = O(h 3 ). Since the card{s : τsh ∩ Γ ̸= ∅} = O(h −1 ), there exists CΓ > 0, depending on the curvature of Γ such that |L h | ≤ CΓ h 2 .

(3.5)

h h Next, consider an element τsh such that τ˚sh ∩ Γ ̸= ∅. Then Γ divides τsh into τs,1 and τs,2 given by h τs,1 := τsh ∩ Ω 1 ,

h τs,2 := τsh ∩ Ω 2 .

(3.6)

Recall that from the definition, the perturbed interface Γh may also intersect τ˚sh and will divide τsh into T τs,h := τsh ∩ Ω 1,h ,

Q τs,h := τsh ∩ Ω 2,h .

(3.7)

Q T Without loss of generality, suppose τs,h is a triangle and τs,h is a quadrilateral; see Fig. 2a, b. Since τsh satisfies the T T h minimal angle condition and one of the angles of τs coincides with an angle of τs,h , it is clear that the triangle τs,h Q will satisfy the maximal angle condition. Now we divide the quadrilateral τs,h using one of its diagonals into two Q,1 Q,2 Q,1 Q,2 Q,1 Q,2 T triangles τs,h and τs,h such that τs,h , τs,h satisfy the maximal angle condition. It is clear that τ˚s,h , τ˚s,h , and τ˚s,h are non-intersecting and Q T τsh = τs,h ∪ τs,h ,

Q Q,1 Q,2 τs,h = τs,h ∪ τs,h ,

for s ∈ E h and τ˚sh ∩ Γ ̸= ∅.

(3.8)

Q,1 Q,2 Q T T We note that the maximal angle condition in τs,h , τs,h , τs,h hold even when τs,h or τs,h are “almost degenerate”, h i.e., when Γh is close to a side or a vertex of the element τs . We now define the continuous and piecewise linear function V˜h on Ω as follows:

• For s ∈ E h , suppose τ˚sh ∩ Γh = ∅. Then ⏐ ⏐ V˜h ⏐τ h := Ih (χ V )⏐τ h , s

s

(3.9)

where Ih (χ V ) is the standard piecewise linear interpolant of χ V , subordinate to the finite element mesh Th .

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• For s ∈ E h , suppose τ˚sh ∩ Γh ̸= ∅, We define ⏐ V˜h ⏐τ T := linear interpolant of χ V s,h

T based on the vertices of the triangle τs,h ;

⏐ V˜h ⏐

Q,1

τs,h

(3.10)

:= linear interpolant of χ V Q,1 based on the vertices of the triangle τs,h ;

(3.11)

⏐ V˜h ⏐τ Q,2 := linear interpolant of χ V s,h

Q,2 based on the vertices of the triangle τs,h .

(3.12)

It is clear from the construction that V˜h is continuous and piecewise linear on Ω . However, V˜h is not piecewise linear subordinate to the mesh Th since V˜h is not linear on the elements τsh that intersect Γh . Remark 3.1. We note that in the construction of V˜h , we have assumed Γh divides τsh into two distinct parts. However if all the points in ∂τsh ∩ Γ are on the same side (face) of τsh , then Γh ∩ τsh = ∅ and Γh does not divide τsh into two distinct parts. In that situation, V˜h |τsh = Ih (χ V )|τsh , i.e., V˜h is the standard linear interpolant of χ V on τsh based on the vertices of τsh . □ The function V˜h for straight Γ : In this case, the perturbation of Γ is not needed and V˜h is defined as in (3.9)–(3.12) with Γh = Γ , T τs,h

=

h τs,1 ,

Ω1,h = Ω1 , Q τs,h

=

Ω2,h = Ω2 ,

h τs,2 ,

for τ˚sh ∩ Γ ̸= ∅

⏐ ⏐ (see (3.3), (3.6), and (3.7)). Thus in the case of straight interface, V˜h ⏐τ h = V ⏐τ h for τs ∩ Γ = ∅. We further note s

s

Q Q,1 Q,2 h that though one does not require to divide τs,h = τs,2 into τs,h , τs,h , as in (3.8), we keep it for efficient numerical integration.

The enrichment space S2h : In the following description, we assume that Γ is curved. The situation for straight interface Γ is simpler and is obtained by simply considering Γh = Γ in the following description. We first set ϕ h (x) = V˜h (x) − Ih V˜h (x), x ∈ Ω , where Ih V˜h is the standard finite element piecewise linear interpolant of V˜h . Then for i ∈ Ih , let ϕih (x) := ϕ h (x),

x ∈ ωih .

(3.13)

It is clear from the definition of V˜h that ϕ h |τs ≡ 0,

if τ˚sh ∩ Γh = ∅.

(3.14)

Also from the definition of V˜h , we have ϕ |τsh ≡ 0 if sides of τsh . Consequently, h

ϕih (x) ≡ 0,

∂τsh

∩ Γh ̸= ∅ and

τ˚sh

∩ Γh = ∅, i.e., when Γh touches one of the

for i ̸∈ Ih,enr ⊂ Ih ,

where Ih,enr := {i ∈ Ih : ωih ∩ Γh ̸= ∅}. We assume that h is small enough such that ⋃ ωih ⊂ Ω0,0 .

(3.15)

(3.16)

i∈Ih,enr

Thus χ(x) = 1 for x ∈ ∪i∈Ih,enr ωih . As a consequence, for an element τsh ∈ ∪i∈Ih,enr ωih , the definition of V˜h does not ⏐ ⏐ depend on the choice of the smooth cutoff function χ, in fact, V˜h τ h is defined by (3.9)–(3.12) with χ replaced by 1. s Thus ϕih , i ∈ Ih,enr , is independent of the choice of χ.


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Since ωih is open, note that ωih ∩ Γh ̸= ∅ implies that there is at least one element τsh , s ∈ E hi , i.e., τsh ⊂ ωih , such that τ˚sh ∩ Γh ̸= ∅. The function ϕih , i ∈ Ih,enr , defined on the patch ωih , is called the enrichment function associated with the node xih , i ∈ Ih,enr . The nodes {xih : i ∈ Ih,enr } are called the enriched nodes in the literature. Furthermore, for i ∈ Ih,enr , if τ˚sh ⊂ ωih does not intersect Γh , then from (3.13), (3.14), we have where τsh ⊂ ωih and τ˚sh ∩ Γh = ∅.

ϕih |τsh = 0,

(3.17)

Thus the enrichment ϕih is non-zero only on those elements in ωih , whose interiors intersect the perturbed interface Γh . Based on the definition of the enrichment function ϕih , i ∈ Ih,enr , the enrichment space S2h is defined as S2h = span{Nih ϕih , i ∈ Ih,enr } = span{Nih (V˜h − Ih V˜h ), i ∈ Ih,enr }. The approximation space for the GFEM is defined as SG F E M = S1h ⊕ S2h { } ∑ ∑ := v ∈ H 1 (Ω ) : v = αi Nih + βi Nih ϕih ; αi , βi ∈ R i∈Ih

{

= v ∈ H (Ω ) : v = 1

∑

i∈Ih,enr

αi Nih

+

i∈Ih

} βi Nih (V˜h − Ih V˜h ) ; αi , βi ∈ R .

∑

(3.18)

i∈Ih,enr

∑ ∑ We note that i∈Ih αi Nih ∈ S1h and i∈Ih,enr βi Nih (V˜h − Ih V˜h ) ∈ S2h . Furthermore, recall that ωih ⊂ Ω0,0 for i ∈ Ih,enr ⏐ (see (3.16)) and therefore χ ⏐ω = 1 for i ∈ Ih,enr and consequently, S2h is independent of the choice of the smooth i cutoff function χ. Clearly, SG F E M ⊂ E(Ω ). Remark 3.2. We note that in the light of the assumption that ∪i∈Ih,enr ωih ⊂ Ω0,0 , we could have defined the function ϕih (s) for i ∈ Ih,enr directly, without introducing the cut-off function χ (since χ = 1 in Ω0,0 ). However, we used χ in the definition of ϕ h as this framework will be used later in the analysis in this paper. □ We now discuss certain notions and properties related to the enrichment function which will be used later in the paper. We define E h,enr := {s ∈ E h : τ˚sh ∩ Γh ̸= ∅} which will be used later in the paper. The elements τsh , s ∈ E h,enr are referred to as the enriched elements. Every vertex (node) xih , i ∈ Ihs of the enriched element τsh is enriched, i.e., for s ∈ E h,enr , we have Ihs ⊂ Ih,enr . We further note that an element τsh , s ∈ E h,enr could be viewed as τsh = ∩i∈Ihs ωih and it is clear from the definition of enrichment function that for s ∈ E h,enr , the enrichment functions associated with ⏐ the vertices of τsh , i.e., ϕih ⏐τ h , i ∈ Ish , are same. We set s ⏐ h,s h⏐ ϕ (x) := ϕi τ h , for all x ∈ τsh where s ∈ E h,enr and i ∈ Ihs . (3.19) s

h h Let i = 1, 2, 3 be the vertices of τsh , s ∈ E h,enr and suppose Γh does not intersect the side [x2,s , x3,s ] of τsh . Then h,s it is clear from the definition of ϕ that for s ∈ E h,enr , we have ⏐ h ϕ h,s (xi,s ) = 0, i = 1, 2, 3, and ϕ h,s ⏐[x h ,x h ] ≡ 0. (3.20) h xi,s ,

2,s

3,s

The Generalized Finite Element Method (GFEM): We first define a perturbation ah (x) of the function a(x) relative to the perturbed interface given by ah (x) = ai ,

x ∈ Ωi,h , i = 1, 2,

where ai ∈ R, i = 1, 2 has been defined after (2.3). Clearly ah (x) depends on h and ah (x) = a(x),

for x ̸∈ L h

where L h was defined in (3.4). We further note that for the straight interface, ah (x) = a(x), x ∈ Ω .

(3.21)

66


The GFEM to approximate the solution of (2.4) is given by u h ∈ SG F E M Bh (u h , v) = F(v),

for all v ∈ SG F E M

(3.22)

where Bh (u, v) :=

∫ (3.23)

ah ∇u · ∇v d x. Ω

We note that Bh (·, ·) is a perturbation of B(·, ·) and depends on h. For the straight interface, Bh (·, ·) = B(·, ·), i.e., there is no perturbation. Remark 3.3. The solution u h of (3.22) is unique up to a constant. To obtain a unique solution, we impose u h (x ∗ ) = 0 where x ∗ ∈ ∂Ω , possibly a boundary node x ∗ = xi∗ , such that exact solution u ∈ E(Ω ) is smooth in a neighborhood of xi∗ . To set v(xi∗ ) = 0 for all v ∈ SG F E M , we set Ih′ = Ih \ xi∗ and consider SG F E M = S1h′ ⊕ S2h , where S1h′ = span{Ni , i ∈ Ih′ } and S2 same as before. However, in the rest of the paper, we will use the notation S1h for S1h′ in the definition of SG F E M with the understanding that v(xi∗ ) = 0 for all v ∈ SG F E M . With this understanding, the stiffness matrix of (3.22) is non-singular and the solution u h is unique. □ Remark 3.4. It is well known (see [33]) that imposing the homogeneous Dirichlet boundary condition only at one point of the functions in the trial space, as we have suggested in Remark 3.3, leads to an increase in the condition number of the associated finite element stiffness matrix. Since the growth in the condition number due to this condition is only logarithmic for the 2D problems that we consider in this paper, we neglect this growth in this paper. But the growth is O(h −1 ) in 3D problems and it has to be addressed. It is possible to avoid this growth by putting more complex constraint on u h ; we refer to [34]. □ Following the notations set at the end of Section 2, we will set Bh,A (·, ·), Eh (A), ∥ · ∥Eh

similar to

B A (·, ·), E(A), ∥ · ∥E , respectively

for any A ⊂ Ω with only a(x) replaced by ah (x). Also we will use Eh := Eh (Ω ). We will show in this paper that the GFEM (3.22) is a Stable Generalized Finite Element Method (SGFEM), i.e., – u h approximates the solution u of (2.4) optimally, i.e., ∥u − u h ∥E(Ω) = O(h); – the conditioning of the GFEM (3.22), with a diagonal scaling to be defined later, is not worse than that of the standard FEM; and – the conditioning of the GFEM (3.22), with the diagonal scaling, is robust with respect to the position of the mesh relative to the interface Γ . Furthermore, we will show the SGFEM, for the interface problem, is also strongly stable – a notion that we will define later. The notion of strong stability yields efficient iterative schemes to solve the underlying linear system of the SGFEM. Remark 3.5. In the standard GFEM, the enrichment functions ϕih are defined as ϕih = ϕ h = V (x) and S2h is defined with Ih,enr = Ih or a suitably chosen Ih,enr ⊂ Ih such that the enriched nodes xi ∈ Ih,enr are at a fixed distance (independent from h) from Γ . On the other hand in the GFEM presented in this paper, we use ϕih as in (3.13) and Ih,enr as in (3.15); consequently |Ih,enr | ≪ |Ih |, the enriched nodes xih , i ∈ Ih,enr are close to Γ , and ϕih (x hj ) = 0 for all nodes x hj ∈ ωih . This choice of enrichment functions and the enriched nodes results into a strongly stable and robust GFEM, i.e., an SGFEM, in terms of the order of convergence, conditioning, and the location of the interface Γ relative to the nodes or the edges of the mesh. Moreover, it requires less number of degrees of freedom. □ We note that the enrichment functions similar to ϕih (based on V instead of Vh ), as defined in (3.13), were introduced in [35]. However, the conditioning on GFEM was not addressed in that paper.


67

For notational convenience, we will not use h in various notations in the rest of the paper; we will use xi , τs , ωi , Ni , ϕ, ϕi , ϕ s τsT , τsQ , τsQ,1 , τsQ,2 S1 , S2

for

xih , τsh , ωih , Nih , ϕ h , ϕih , ϕ h,s , respectively,

for

Q Q,1 Q,2 T τs,h , τs,h , τs,h , τs,h respectively,

for

S1h , S2h respectively,

with the understanding that they depend on the discretization parameter h. 4. The optimal convergence of the GFEM solution In this section, we show that the solution u h ∈ SG F E M of (3.22) converges optimally to the exact solution u ∈ E of (2.4). This will be done by considering a perturbed problem, namely, u(h) ∈ Eh = {w ∈ H 1 (Ω ) : ∥w∥Eh < ∞}, Bh (u(h), v) = F(v), for all v ∈ Eh ,

(4.1)

where Bh (u, v) is given in (3.23). Note that Bh (u, v) is defined in terms of the perturbed function ah and therefore the solution u(h) ∈ E depends on h. We particularly note that u(h) does not belong to a finite dimensional space and it is a perturbation of the exact solution u ∈ E of the main problem (2.4). Furthermore, it is easy to show that β1 β0 Bh (w, w) ≤ B(w, w) ≤ Bh (w, w), for all w ∈ H 1 (Ω ). (4.2) β1 β0 Therefore the norms ∥ · ∥E and ∥ · ∥Eh are equivalent, u(h) ∈ E, and Bh (u(h), v) = F(v),

for all v ∈ E.

(4.3)

We first show that u(h) and u are “close”. Lemma 4.1. Let u and u(h) be the solutions of the problems (2.4) and (4.1) respectively. Then there exists C > 0, independent of h, such that ∥u − u(h)∥E ≤ Ch|u|1,∞,Ω . Proof. From the definition of u and the fact that u(h) satisfies (4.3), we have B(u, v) = F(v) = Bh (u(h), v),

for all v ∈ E.

Therefore ∫ (a − ah )∇u · ∇v d x = B(u, v) − Bh (u, v) = Bh (u(h), v) − Bh (u, v) Ω

= Bh (u(h) − u, v),

for all v ∈ E.

Substituting v = u(h) − u and using (4.2) and Schwarz inequality, we get β0 ∥u(h) − u∥2E ≤ Bh (u(h) − u, u(h) − u) β1 ∫ (a − ah )∇u · ∇(u(h) − u) d x

= Ω

≤

1 ∥(a − ah )∇u∥0,Ω ∥u(h) − u∥E β0

and therefore, ∥u(h) − u∥E ≤

β1 ∥(a − ah )∇u∥0,Ω . β02

(4.4)

Now using (3.21) and (3.5), we have ∫ ∫ ∥(a − ah )∇u∥20,Ω = (a − ah )2 |∇u|2 d x = (a − ah )2 |∇u|2 d x Ω

Lh

≤ CΓ h 2 |a1 − a2 |2 |u|21,∞,Ω

68


Therefore from (4.4), we get the desired result ∥u(h) − u∥E ≤ Ch|u|1,∞,Ω √ where C = ββ12 |a1 − a2 | CΓ , i.e., C is independent of h.

■

0

We next prove a technical result that we will use later. Lemma 4.2. For an element τs , s ∈ E h such that τ˚s ∩ Γh ̸= ∅, there exists C > 0, independent of h and s, such that ∥χ V − V˜h ∥1,τs ≤ Ch 3/2 V, where V is defined in (2.8). ⏐ ⏐ Proof. Since τ˚s ∩ Γh ̸= ∅, we first note from (3.16) that τs ⊂ Ω0,0 and χ|τs = 1. Therefore χ V ⏐τs = V ⏐τs . Moreover, ⏐ as mentioned before, V˜h ⏐τs for τ˚s intersecting Γh does not depend on the smooth cutoff function χ. Recall (see (3.7)) that Γh divides τs into τsT and τsQ = τsQ,1 ∪ τsQ,2 . In this case, Γ also divides τs into τs,1 and τs,2 T Q such that τ˚s,1 ⊂ Ω1 and τ˚s,2 ⏐ ⊂ Ω2 (see (3.6)). We first assume that τs,1 ⊂ τs and τs ⊂ τs,2 ; see Fig. 2. From the ⏐ definition of V , we have V Γ = 0 and V ⏐ is smooth on τs,1 , τs,2 . Let V be the smooth extension of V ⏐τ to τsT such that V = V on τs,1 and s,1

∥V ∥W∞ ℓ (τ T ) ≤ C∥V ∥W ℓ (τ ) , s ∞ s,1

ℓ = 1, 2.

(4.5) ⏐ Furthermore, V = V at all the three vertices of τsT . Recalling that V˜h ⏐τ T is the linear interpolant of V on τsT (note s ⏐ χ = 1), we infer that V˜h ⏐τ T is also the linear interpolant of V on τsT . Therefore, using (4.5), s

∥V − V˜h ∥1,τsT ≤ ∥V − V˜h ∥1,τsT + ∥V − V ∥1,τsT ≤ Ch 2 |V |W∞ 2 (τ T ) + ∥V − V ∥1,τ T s s ≤ Ch 2 |V |W∞ 2 (τ ) + ∥V − V ∥1,τ T , s s,1

(4.6)

where we have used the standard interpolation result ([36], Theorem 3.1.5). ⏐ Let L s := τsT \ τ˚s,1 . Clearly, |L s | = O(h 3 ). Since V ⏐Γ = 0, V = V on τs,1 and L s = τsT ∩ τs,2 , we have 2

2

2

2 ∥V − V ∥1,τsT = ∥V − V ∥1,L s ≤ C[∥V ∥W∞ 1 (τ T ) + ∥V ∥ 1 s W (τ

∞ s,2 )

≤ C[∥V ∥2W 1 (τ

∞ s,1 )

+ ∥V ∥2W 1 (τ

] h3

∞ s,2 )

] h3,

where we used (4.5) with ℓ = 1. Therefore from (4.5) ∥V − V˜h ∥1,τsT ≤ Ch 3/2 [∥V ∥2W 2 (τ

∞ s,1 )

+ ∥V ∥2W 1 (τ

∞ s,2 )

]1/2 .

(4.7)

Now, since V˜h is the linear interpolant of V on τsQ,i ⊂ τsQ ⊂ τs,2 for i = 1, 2, using a standard interpolation result we have ∥V − V˜h ∥1,τ Q,i ≤ Ch 2 |V |W 2 (τ Q ) , i = 1, 2, s

∞ s

and therefore from (4.6) and (4.7) ∥V − V˜h ∥1,τs ≤ Ch 3/2 [∥V ∥2W 2 (τ

∞ s,1 )

+ ∥V ∥2W 1 (τ

∞ s,2 )

]1/2

+ Ch 2 |V |W 2 (τ Q ) ∞ s

≤ Ch 3/2 V, which is the desired result when τs,1 ⊂ τsT and τsQ ⊂ τs,2 . On the other hand, suppose τsT ⊂ τs,1 and τs,2 ⊂ τsQ . Then we consider the smooth extension V of V |τs,2 to τsQ and repeat similar arguments as above to get the desired result. Finally, if Γ is on both sides of Γh , i.e., Γ intersects Γh in τ˚s , then we have to consider smooth extensions of both V |τs,1 and V |τs,2 and repeat similar arguments to get the desired result. ■


69

The following Lemma plays a crucial role in proving the optimal convergence. Lemma 4.3. Let u be the solution of (2.4) and assume it is of the form (2.5). Then there exists u ∗h ∈ SG F E M such that ∥u − u ∗h ∥1,Ω ≤ Ch{ |u 0 |2,Ω + ∥χ H ∥W∞ 2 (Ω ) V }, 0

(4.8)

where C > 0 is independent of h. Proof. Recall that the exact solution u is of the form (see (2.5)) u = u 0 + AV, where A = χ H ; note A is smooth with support Ω 0 . We consider u ∗h ∈ S SG F E M defined as (see (3.18)) ∑[ ∑ ] u ∗h = u 0 (xi ) + A(xi )V˜h (xi ) Ni + A(xi )Ni (V˜h − Ih V˜h ). i∈Ih

(4.9)

i∈Ih,enr

Note that ∑[ ] u 0 (xi ) + A(xi )V˜h (xi ) Ni ∈ S1h , i∈Ih

∑

A(xi )Ni (V˜h − Ih V˜h ) ∈ S2h .

i∈Ih,enr

⏐ ⏐ From the definition of V˜h , we have (V˜h − Ih V˜h )⏐ω = 0 for i ̸= Ih,enr . Moreover, (V˜h − Ih V˜h )⏐τs = 0 for i ⏐ τs ⊂ ωi , i ∈ Ih,enr such that τ˚s ∩ Γh = ∅. Thus (V˜h − Ih V˜h )⏐τs ̸≡ 0 only when τ˚s ∩ Γh ̸= ∅. We first consider the elements τs such that τ˚s ∩ Γh ̸= ∅. We write u = u 0 + AV = u 0 + A V˜h + A(V − V˜h ).

(4.10)

From Lemma 4.2, we have (recall χ = 1 on τs ) [ 2 2 2 ] ∥A(V − V˜h )∥1,τs ≤ C |A|2W 1 (τ ) ∥V − V˜h ∥0,τs + ∥A∥2L ∞ (τs ) |V − V˜h |1,τs ∞ s

≤

C∥A∥2W 1 (Ω ) ∥V ∞ 0

2 − V˜h ∥1,τs ≤ Ch 3 ∥A∥2W 1 (Ω ) V2 . ∞

(4.11)

0

Next we note that for τ˚s ∩ Γh ̸= ∅, we have Ihs ⊂ Ih,enr . Also ∑ v(xi )Ni = Ih v|τs i∈Ihs

for a smooth function v. We further note that qs := [(Ih A)(Ih V˜h )] |τs is quadratic on τs with [(Ih A)(Ih V˜h )](xi ) = A(xi )V˜h (xi ) for i ∈ Ihs . Therefore (for τ˚s ∩ Γh ̸= ∅) ∑[ ∑ ⏐ ] ⏐ u 0 (xi ) + A(xi )V˜h (xi ) Ni ⏐τs + A(xi )Ni (V˜h − Ih V˜h )⏐τs u ∗h |τs = i∈Ihs

i∈Ihs

= Ih u 0 |τs + Ih [(Ih A)(Ih V˜h )] |τs + (V˜h − Ih V˜h )|τs

∑

A(xi )Ni |τs

i∈Ihs

= Ih u 0 |τs + Ih [(Ih A)(Ih V˜h )] |τs + V˜h (Ih A)|τs − (Ih V˜h )(Ih A)|τs = Ih u 0 |τs + V˜h (Ih A)|τs − qs + Ih qs |τs . Thus ⏐ ⏐ [ ]⏐ [ ] ⏐ (u 0 + A V˜h − u ∗h )⏐τs = u 0 − Ih u 0 ⏐τs + A − Ih A V˜h ⏐τs + (qs − Ih qs )⏐τs .

(4.12)

Now using the standard interpolation estimate, we have ∥qs − Ih qs ∥1,τs ≤ Ch 2 |qs |W∞ 2 (τ ) . s Since qs is quadratic, using the equivalence of norms and boundedness of V˜h , we get ∥qs − Ih qs ∥1,τs ≤ Ch 2 ∥A∥ L ∞ (τs ) ∥V˜h ∥ L ∞ (τs ) ≤ Ch 2 ∥A∥ L ∞ (Ω0 ) V.

(4.13)

70


Next using ∥V˜h ∥W∞ 1 (τ ) ≤ C∥V ∥W 1 (τ ) and the standard interpolation result, we have s ∞ s ∥[A − Ih A]V˜h ∥1,τs ≤ Ch 2 ∥A∥W∞ 2 (Ω ) V, 0 ∥u 0 − Ih u 0 ∥1,τs ≤ Ch|u 0 |2,τs , and therefore using (4.10), (4.11), (4.12), and (4.13), we get ∥u − u ∗h ∥1,τs ≤ ∥u 0 + A V˜h − u ∗h ∥1,τs + ∥A(V − V˜h )∥1,τs 3/2 ∥A∥W∞ ≤ Ch|u 0 |2,τs + Ch 2 ∥A∥W∞ 1 (Ω ) V 2 (Ω ) V + Ch 0 0

≤ Ch|u 0 |2,τs + Ch 3/2 ∥A∥W∞ 2 (Ω ) V, 0

for τ˚s ∩ Γ ̸= ∅.

(4.14)

We now consider the element τs such that τs ∩ Γ = ∅. From (4.5) we have ∑[ ⏐ ⏐ ] u ∗h |τs = u 0 (xi ) + A(xi )V˜h (xi ) Ni |τs = Ih u 0 ⏐ + Ih (AV )⏐ , τs

i∈Ihs

τs

⏐ since V˜h (xi ) = V (xi ) for i ∈ Ihs and (V˜h − Ih V˜h )⏐τs ≡ 0. Therefore using the standard interpolation result we have ∥u − u ∗h ∥1,τs ≤ ∥u 0 − Ih u 0 ∥1,τs + ∥AV − Ih (AV )∥1,τs for τs ∩ Γ = ∅.

≤ Ch|u 0 |2,τs + Ch|AV |2,τs

Thus, from (4.14) and above, we have ∑ ∑ ∥u − u ∗h ∥21,Ω = ∥u − u ∗h ∥21,τs + ∥u − u ∗h ∥21,τs τ˚s ∩Γ =∅ 2

≤ Ch [

τ˚s ∩Γ ̸=∅

∑

{|u 0 |22,τs

∑

+ |AV |22,τs } ] + Ch 2

τ˚s ∩Γ =∅

|u 0 |22,τs

τ˚s ∩Γ ̸=∅

+ Ch

3

∑ τ˚s ∩Γ ̸=∅

∥A∥2W 2 (Ω ) V2 ∞ 0

≤ Ch 2 { |u 0 |22,Ω + |AV |22,Ω0 } + Ch 2 ∥A∥2W 2 (Ω ) V2 ∞

0

≤ Ch 2 { |u 0 |22,Ω + ∥A∥2W 2 (Ω ) V2 }, ∞

0

where we used the fact that card{s : τ˚s ∩ Γ ̸= ∅} = O(h −1 ). Thus recalling that A = χ H , we obtain the desired result. ■ Remark 4.1. We note that we do not have the Galerkin orthogonality, namely, B(u − u h , v) ̸= 0,

for all v ∈ SG F E M ,

where u h ∈ SG F E M is the solution of (3.22). Therefore we cannot infer that u h is the best approximation of u in the energy norm; consequently, Lemma 4.3 does not yield the desired optimal convergence result. We note however that u h satisfies the orthogonality ( ) Bh u(h) − u h , v = 0, for all v ∈ SG F E M , (4.15) which will be used in the next result. □ We now prove the main convergence result. Theorem 4.4. Suppose u ∈ E(Ω ) and u h ∈ SG F E M be the solutions of the problems (2.4) and (3.22) respectively. Then there exists C > 0, independent of h, such that ∥u − u h ∥E ≤ Ch{|u|1,∞,Ω + |u 0 |2,Ω + ∥χ H ∥W∞ 2 (Ω ) V}. 0


71

Proof. Recall u(h) is the solution of the perturbed problem (4.1). Then using (4.15) we have, ∥u − u h ∥E ≤ ∥u − u(h)∥E + ∥u(h) − u h ∥E ≤ ∥u − u(h)∥E + C inf ∥u(h) − v∥E v∈SG F E M

≤ ∥u − u(h)∥E + C∥u(h) − u ∗h ∥E

(4.16)

where u ∗h ∈ SG F E M satisfies (4.8) in Lemma 4.3. We note that C does not depend on h. Now using (4.8) and Lemma 4.1, we have ∥u(h) − u ∗h ∥E(Ω) ≤ ∥u(h) − u∥E + ∥u − u ∗h ∥E ≤ Ch|u|1,∞,Ω + Ch{|u 0 |2,Ω + ∥χ H ∥W∞ 2 (Ω ) V} 0 and therefore using Lemma 4.1 in (4.16), we have ∥u − u h ∥E ≤ Ch{|u|1,∞,Ω + |u 0 |2,Ω + ∥χ H ∥W∞ 2 (Ω ) V} 0 which is the desired result. ■ Remark 4.2. We have provided the convergence analysis of the curved interface in this section. For the straight interface, the analysis is much simpler and we do not provide it here. We note that Lemma 4.1 is trivially true as u = u(h) for the straight interface as we do not need perturbation of Γ . The proofs of Lemmas 4.2 and 4.3 are much simpler and could be derived following the same ideas. Finally Theorem 4.4 could be obtained using directly the Galerkin orthogonality, which holds for the straight interface. □ 5. Sufficient conditions for the well-conditioning of the GFEM In this section, we will present sufficient conditions on the enrichment space S2 that will ensure that the GFEM (3.22) is well-conditioned, i.e., the conditioning of the GFEM is not worse than the conditioning of the standard FEM. Moreover, we will also define the notion of a strongly stable GFEM — strong stability of the GFEM facilitates efficient implementation of iterative methods to solve the underlying linear system. Later in the paper, we will prove that these sufficient conditions hold for the particular enrichment space S2 that we have considered for the interface problems in this paper. Recall that S1 = span{Ni , i ∈ Ih } and S2 = span{Nl ϕl , l ∈ Ih,enr }, where Ni is the hat function associated with the node xi and ϕl = (V˜h − Ih V˜h )|ωl is the enrichment function associated with the enriched node xl ; ωl is the patch associated with xl . The stiffness matrix of the GFEM, associated with the approximation space SG F E M = S1 ⊕ S2 , is of the form [ ] A A12 A = 11 (5.1) A21 A22 where A11 = {Bh (N j , Ni )}i, j∈Ih ,

A22 = {Bh (Nl ϕl , Nk ϕk )}l,k∈Ih,enr

are the stiffness matrices associated with S1 and S2 respectively, while T = {Bh (Nl ϕl , Ni )}l∈Ih,enr ,i∈Ih . A12 = A21

Clearly, A11 is |Ih | × |Ih | and A22 is |Ih,enr | × |Ih,enr |. We note that A11 is the standard finite element matrix associated with the finite element triangulation used in the definition of SG F E M . Moreover, A11 is non-singular based on the redefinition of Ih elaborated in Remark 3.3. We note that for the straight interface, we have Bh (·, ·) = B(·, ·). Next, we define the scaled stiffness matrix [ ] Aˆ Aˆ Aˆ = DAD = ˆ 11 ˆ 12 (5.2) A21 A22 where D = diag{D11 , D22 }

72


with D11 , D22 diagonal matrices given by −1/2

(D11 )ii = (A11 )ii

−1/2

and (D22 )kk = (A22 )kk .

ˆ Aˆ 11 = D11 A11 D11 , and Aˆ 22 = D22 A22 D22 are equal to 1. Also A ˆ 21 = Aˆ T . The Clearly, the diagonal elements of A, 12 scaled condition number of A and A11 is given by ˆ and K(A11 ) := κ2 (Aˆ 11 ) K(A) := κ2 (A) where κ2 (·) is the condition number subordinate to the Euclidean vector norm ∥ · ∥. The GFEM is well-conditioned if K(A) ≈ K(A11 ). The scaled stiffness matrix Aˆ could be interpreted as the stiffness matrix associated with the scaled shape functions ℵi = µi Ni for i ∈ Ih ,

Ψk = νk Nk ϕk for k ∈ Ih,enr

(5.3)

where µi , νk ∈ R such that ∥ℵi ∥2Eh := Bh (ℵi , ℵi ) = 1,

∥Ψk ∥2Eh := Bh (Ψk , Ψk ) = 1.

It is clear that ℵi , Ψk have compact supports ωi , ωk respectively. Also S1 = span{ℵi , i ∈ Ih } and S2 = span{Ψk , k ∈ Ih,enr }. Now using the definition of the diagonal matrix D, it is easy to see that Aˆ 11 = {Bh (ℵi , ℵ j )}i, j∈Ih , Aˆ 22 = {Bh (Ψl , Ψk )}l,k∈Ih,enr T Aˆ 12 = Aˆ 21 = {Bh (Ψl , ℵi )}l∈Ih,enr , i∈Ih .

(5.4)

We also define the scaled element matrices associated with an enriched element τs and the space S2 given by { } Aˆ (s) s ∈ E h,enr . (5.5) 22 := Bh,τs (Ψl , Ψk ) l,k∈I s , h

We assume that the enrichment space S2 satisfy the following axioms: Axiom A1. The space S2 is almost orthogonal to the space S1 with respect to the inner product Bh (·, ·), i.e., there exist positive constants L 1 and U1 , independent of h, such that [ ] [ ] L 1 ∥v1 ∥2E + ∥v2 ∥2E ≤ Bh (v1 + v2 , v1 + v2 ) ≤ U1 ∥v1 ∥2E + ∥v2 ∥2E (5.6) for all v1 ∈ S1 and v2 ∈ S2 . It is possible to show (see Theorem 3.4 in [31]) that if S2 satisfies Axiom A1, then K(A) ≈ max{K(A11 ), K(A22 )}. Thus if K(A22 ) ≈ K(A11 ), then it is immediate that K(A) ≈ K(A11 ) i.e., the conditioning of the GFEM is not worse than that of the standard FEM and the GFEM is a Stable GFEM (SGFEM). We now define the notion of a Strongly Stable GFEM. A GFEM is strongly stable if the enrichment space S2 , in addition to Axiom A1, also satisfies the following axiom: Axiom A2. There exist positive constants L 2 and U2 , independent of h, such that −1 2 2 T L 2 ∥D−1 22 x∥ ≤ x A22 x ≤ U2 ∥D22 x∥ ,

for all x ∈ R|Ih,enr | .

(5.7)

For completeness, we mention following result was proved in [28,31]. Theorem 5.1. If the spaces S1 and S2 satisfy the conditions A1 and A2, then the scaled condition number K(A) of A is of the same order as the scaled condition number K(A11 ), in particular, U1 max[1, U2 /λmax (Aˆ 11 )] L1 K(A11 ) ≤ K(A) ≤ K(A11 ) U1 L 1 min[1, L 2 /λmin (Aˆ 11 )] where λmin (Aˆ 11 ), λmax (Aˆ 11 ) are the smallest and largest eigenvalues of Aˆ 11 . ■


73

We mention that a GFEM, with enrichment spaces satisfying the additional Axiom A2, yields efficient iterative methods to solve the underlying linear system; it was numerically established for the enrichment space S2 (considered in this paper) for interface problems in [32]. We first present a result that will be used to establish (5.6). For s ∈ E h,enr , set ⏐ S2,s := {v ⏐τs : v ∈ S2 }. Let Πs : H 1 (τs ) → S2,s be the H 1 -projection, namely (w − Πs w, v)τs := (w − Πs w, v) H 1 (τs ) ∫ = ∇(w − Πs w) · ∇v d x = 0, τs

for all v ∈ S2,s .

⏐ Also set S1,s := {w ⏐τ h : w ∈ S1 }. Clearly, S1,s = P 1 (τs ), where P 1 (τs ) is the space of linear polynomials on τs . s

Lemma 5.2. Suppose there exists 0 < C ≤ 1, independent of s, h, and the position of Γh in τs , such that |(I − Πs )v|1,τs ≥ C, |v|1,τs

for all v ∈ S1,s = P 1 (τs ).

(5.8)

Then there exist constants L 1 and U1 , independent of h and the position of Γh relative to the mesh, such that [ ] [ ] L 1 ∥v1 ∥2E + ∥v2 ∥2E ≤ Bh (v1 + v2 , v1 + v2 ) ≤ U1 ∥v1 ∥2E + ∥v2 ∥2E , for all v1 ∈ S1 and v2 ∈ S2 . Proof. We first consider 0 < C < 1. Let v1 ∈ S1 and v2 ∈ S2 . On τs , we write v1 |τs = (I − Πs )v1 |τs + Πs v1 |τs . Since Πs v1 ∈ S2,s , we have ((I − Πs )v1 , Πs v1 )τs = 0 . Therefore using (5.8), we have |v1 |21,τs = (v1 , v1 )τs ( ) = (I − Πs )v1 + Πs v1 , (I − Πs )v1 + Πs v1 τs = |(I − Πs )v1 |21,τs + |Πs v1 |21,τs ≥ C 2 |v1 |21,τs + |Πs v1 |21,τs , and thus |Πs v1 |21,τs ≤ (1 − C 2 )|v1 |21,τs , where 0 < C < 1 by assumption. Now for any ϵ > 0, we have ∫ ⏐ ⏐√ ⏐ ϵ∇v2 · √1 ∇(Πs v1 )⏐ d x 2|(v2 , Πs v1 )τs | = 2 ϵ τs 1 2 ≤ ϵ|v2 |1,τs + |Πs v1 |21,τs ϵ 1 − C2 ≤ ϵ|v2 |21,τs + |v1 |21,τs , ϵ where we used (5.9) in the last step. We choose 0 < ϵ < 1 such that { 1 − C2 } max ϵ, := C ∗ < 1 . ϵ

(5.9)

74


Thus we have [ ] 2|(v2 , Πs v1 )τs | ≤ C ∗ |v1 |21,τs + |v2 |21,τs .

(5.10) ∗

Note that C is independent of s and h and therefore C , which depends only on C, is independent of s, h, and the position of Γh in τs . Next, since v2 |τs ∈ S2,s , from the definition of Πs , we have (v1 , v2 )τs = ((I − Πs )v1 + Πs v1 , v2 )τs = (Πs v1 , v2 )τs , and therefore using (5.10), (v1 + v2 , v1 + v2 )τs = |v1 |21,τs + |v2 |21,τs + 2(v1 , v2 )τs = |v1 |21,τs + |v2 |21,τs + 2(Πs v1 , v2 )τs ≥ |v1 |21,τs + |v2 |21,τs − 2|(Πs v1 , v2 )τs | [ ] ≥ |v1 |21,τs + |v2 |21,τs − C ∗ |v1 |21,τs + |v2 |21,τs [ ] = (1 − C ∗ ) |v1 |21,τs + |v2 |21,τs . ⏐ Recall that Ω = ∪s∈E h τs and from the definition of S2 , we have v2 ⏐τs ≡ 0 for s ∈ E h \ E h,enr . Now ∫ ∫ Bh (v1 + v2 , v1 + v2 ) = ah (x)|∇(v1 + v2 )|2 d x ≥ β0 |∇(v1 + v2 )|2 d x Ω Ω [ ∑ ] ∑ = β0 (v1 + v2 , v1 + v2 )τs + (v1 , v1 )τs s∈E h,enr

(5.11)

s∈E h \E h,enr

∑ [ ] ≥ β0 (1 − C ) |v1 |21,τs + |v2 |21,τs + β0

∑

∗

s∈E h,enr

|v1 |21,τs

s∈E h \E h,enr

[ ] ≥ β0 (1 − C ) |v1 |21,Ω + |v2 |21,Ω [ ] β0 ≥ (1 − C ∗ ) ∥v1 ∥2E + ∥v2 ∥2E , β1 ∗

where we have used (2.3) and (5.11). We set L 1 := of Γ relative to the mesh. It is easy to show that [ ] Bh (v1 + v2 , v1 + v2 ) ≤ 2 ∥v1 ∥2E + ∥v2 ∥2E ,

(5.12) β0 (1 β1

− C ∗ ); it is clear the L 1 is independent of h and the position

and therefore, together with (5.12), we have ⏐ the desired result with U1 := 2. We now consider C = 1. Then Πs v1 ⏐τs = 0 and it is easy to see from (5.11) that we get the desired result with L 1 = U1 = 1. ■ We next present a result that will be used to establish (5.7). Lemma 5.3. For each enriched node xi , i ∈ Ih,enr , suppose (a) there exists at least one element τs , s ∈ E hi ∩ E h,enr such that C1 δi2 ≤ δ T Aˆ (s) 22 δ,

∀δ ∈ R3 ,

(5.13)

where δi is the component of δ associated with the node xi of τs and the constant C1 > 0 is independent of i, h, and the position of Γh in τs . (b) for all τs , s ∈ E hi ∩ E h,enr , there exists a constant C2 > 0, independent of i, h, and the position of Γh in τs such that 2 δ T Aˆ (s) 22 δ ≤ C 2 ∥δ∥ ,

∀δ ∈ R3 .

(5.14)

Then there exist L 2 , U2 > 0, independent of h and the position of Γh relative to the mesh such that −1 2 2 T L 2 ∥D−1 22 x∥ ≤ x A22 x ≤ U2 ∥D22 x∥ ,

for all x ∈ R|Ih,enr | .


75

Proof. Let γˆ = (γˆ1 , γˆ2 , . . . , γˆ|Ih,enr | )T ∈ R|Ih,enr | , where each component γˆ j of γˆ is associated with a node x j , j ∈ Ih,enr . We consider τs , s ∈ E hi ∩ E h,enr , and let δs ∈ R3 whose components are those of γˆ , associated with the indices Ihs of the vertices of the element τs . We set δs,i to be the component of δs that is associated with the node xi , i.e., δs,i = γî It is easy to check that ) ∑ ∑ ( ∑ T ˆ δsT Aˆ (s) δsT Aˆ (s) 22 δs = 3 γˆ A22 γˆ . 22 δs ≤ 3 i∈Ih,enr s∈E i ∩E h,enr h

s∈E h,enr

Therefore using (5.13), we have ∑ ( ∑ ˆ 22 γˆ ≥ 1 γˆ T A 3 i∈I i h,enr

δsT Aˆ (s) 22 δs

)

s∈E h ∩E h,enr

C1 ∑ 2 ≥ δs,i 3 i∈I h,enr

C1 C1 ∑ 2 γî = ∥γˆ ∥2 . = 3 i∈I 3

(5.15)

h,enr

Next, γˆ T Aˆ 22 γˆ =

∑

δsT Aˆ (s) 22 δs

s∈E h,enr

≤

∑ (

∑

δsT Aˆ (s) 22 δs

)

i∈Ih,enr s∈E i ∩E h,enr h

≤

∑ (

∑

) C2 ∥δs ∥2 ≤ 3C2 ∥γˆ ∥2 .

(5.16)

i∈Ih,enr s∈E i ∩E h,enr h |Ih,enr | Now substituting γˆ = D−1 in (5.15) and (5.16), and recalling that Aˆ 22 = D22 A22 D22 , we get the 22 x with x ∈ R desired result −1 2 2 T L 2 ∥D−1 22 x∥ ≤ x A22 x ≤ U2 ∥D22 x∥ ,

for all x ∈ R|Ih,enr | ,

with L 2 = C1 /3 and U2 = 3C2 and where C1 , C2 are independent of i, h, and the position of Γh in the enriched elements, i.e., relative to the mesh. ■ It is clear from Lemmas 5.2 and 5.3 that if the enrichment space S2 satisfies the element wise sufficient conditions (5.8), (5.13), and (5.14), then the Axioms A1 and A2 hold, and consequently Theorem 5.1 holds, i.e., the underlying GFEM is strongly stable. Remark 5.1. We note that the sufficient conditions (5.8), (5.13), and (5.14) given in Lemmas 5.2 and 5.3 are quite general in the sense that they could be used to establish Axioms A1 and A2 for a GFEM applied to applications other than the interface problems (where typically a different enrichment space will be used). Of course, one has to show that the constants in these Lemmas are independent of the “features” of the application problem. □ In the rest of the paper, we will show that the enrichment space S2 for the interface problem satisfies Axioms A1 and A2. In particular, we will show that the constants L 1 , U1 , L 2 and U2 in the Axioms are independent of the position of the perturbed interface Γh (thus of the position of Γ ) relative to a node xi , i ∈ Ih,enr or the sides of an element τs , s ∈ E h,enr . This is one of the main features of this paper. 6. Axiom A1 — Almost orthogonality The sufficient condition (5.8) for the Axiom A1 presented in the last section is element based; an important assumption in this sufficient condition is that the associated constants are independent of the position of Γh (and therefore of Γ ) relative to the mesh. In this section we show that the enrichment space S2 indeed satisfies this sufficient condition by presenting the analysis on the master element.

76


Fig. 3. Master Element; Γh |τs is mapped to M N .

Let τs be an element with s ∈ E h,enr , i.e., τ˚s intersects Γh . Then τs = τsT ∪ τsQ,1 ∪ τsQ,2 (see (3.8)). Suppose Fs : T → τs is the affine mapping, where T is the master √equilateral triangle ABC with sides of length 1 and the coordinates of the vertices A, B, C are (0, 0), (1, 0), ( 21 , 23 ), respectively (see Fig. 3). Also suppose Γh in τs is mapped to the line segment M N that intersects T . The line segment M N will intersect either the sides AB, AC, or AB, BC, or AC, BC of T , depending on the position of the Γh in τs . We consider the case when M N intersects the sides AB, AC and M ∈ AB and N ∈ AC. We set ξ := |AM|, η := |AN |, T1 := △AM N = Fs−1 (τsT ), T21 := △N M B = Fs−1 (τsQ,1 ), T22 := △N BC = Fs−1 (τsQ,2 ) . Note that ξ, η, T1 , T21 , T22 depend on the position of Γh in τs ; also T1 , T21 , T22 satisfy the maximal angle condition. Let N A , N B , NC be the usual Lagrange linear functions associated with the vertices A, B, C, respectively, of the triangle T . For the element τs , let S2T := S2T (ξ, η) = {w = v(Fs−1 (τs )) : v ∈ S2 } , where

S2T (ξ, η)

indicates that the functions in

S2T

w = (a N A + bN B + cNC )ϕT = LϕT ,

depend on ξ, η. Any w ∈

(6.1) S2T

could be written as

a, b, c ∈ R,

(6.2)

where L := a N A + bN B + cNC ∈ P 1 (T ), P 1 (T ) is the space of linear polynomials on T , and ϕT := ϕ s (Fs−1 (τs )); ϕ s is the enrichment function on the element τs (see (3.19)). Note that ϕT is continuous in T ; it is linear in T1 , T21 , T22 and it depends on ξ, η. We will refer to ϕT as ϕξ,η := ϕT in this section. Furthermore, from (3.20), we have ϕξ,η (A) = 0 and ϕξ,η = 0 on BC for all ξ, η. Therefore w(A) = 0, w| BC = 0,

for all w ∈ S2T .

We similarly define S1T := {L = u(Fs−1 (τs )) : u ∈ S1 } = P 1 (T ) . Lemma 6.1. There exists C > 0, independent of ξ and η, such that |(I − ΠT )u|1,T ≥ C, for all u ∈ S1T = P 1 (T ) , |u|1,T where ΠT : H 1 (T ) → S2T is the projection defined by ∫ ∇(u − ΠT u) · ∇v d x = 0, for all v ∈ S2T . T

(6.3)


77

Fig. 4. (a) 0 < ξ j ≤ η j ≤ 1/2; (b) 0 < ξ j ≤ η j and 1/2 < η j < 1; (c) |AN ∗j | = |AM j | = ξ j ≤ η j .

Proof. We first prove this result for 0 < ξ ≤ η < 1. For the clarity of notation, we will suppress T and will use S2 instead of S2T . Let S¯1 := {w ∈ P 1 (T ) : |w|1,T = 1} and for u ∈ S¯1 , define W (u) := |(I − ΠT )u|1,T =

inf

v∈S2 (ξ,η)

|u − v|1,T .

Since W (u) = W (u + J ) for any constant J , then (6.3) is equivalent to the statement that there exists C > 0, independent of ξ, η, such that W (u) :=

inf

v∈S2 (ξ,η)

|u − v|1,T ≥ C,

∀u ∈ S1∗ := {u ∈ S¯1 : u(B) = 0}.

(6.4)

We will prove the above statement. Case 1: We first consider the case where 0 < ξ ≤ 1/2 and suppose that (6.4) does not hold. Then, there exist sequences {ξ j }, {η j }, {u j ∈ S1∗ } such that 0 < ξ j ≤ 1/2, ξ j ≤ η j ≤ 1, and inf

v∈S2 (ξ j ,η j )

|u j − v|1,T < 1/j .

Note that for each j, the interface intersects the edges AB, AC at the points M j , N j respectively, where |AM j | = ξ j , |AN j | = η j . For each u j , we choose v j = L j ϕξ j ,η j ∈ S2 (ξ j , η j ), where L j ∈ P 1 (T ), such that |u j − v j |1,T ≤ 2/j .

(6.5)

Note that for T1, j := △AM j N j , T21, j := △N j M j B, and T22, j := △N j BC, we have ⏐ ⏐ ⏐ ⏐ v j ⏐T ∈ P 2 (T1, j ), v j ⏐T ∈ P 2 (T21, j ), v j ⏐T ∈ P 2 (T22, j ), and v j ⏐ BC = 0, 22, j

21, j

1, j

where P (T1, j ), P (T21, j ), P (T22, j ) are the spaces of quadratic polynomials on T1, j , T21, j , T22, j respectively. Next, since |u j |1,T = 1, there exists a subsequence, denoted again by {u j }, that converges in L 2 ; there exists u 0 ∈ S1∗ , i.e., u 0 ∈ P 1 (T ), |u 0 |1,T = 1, u 0 (B) = 0, such that 2

2

2

∥u j − u 0 ∥ L 2 (T ) ≤ C/j , where C is independent of j. Moreover, since u j − u 0 ∈ P 1 (T ) and (u j − u 0 )(B) = 0, using the equivalence of norms and above inequality, we have |u j − u 0 |1,T ≤ C∥u j − u 0 ∥ L 2 (T ) ≤ C/j , and therefore using (6.5), |u 0 − v j |1,T ≤ |u 0 − u j |1,T + |u j − v j |1,T ≤ C/j ,

for j large.

(6.6)

Next let D be mid-point of the edge AB, F be the mid-point AC, and F B intersects C D at E (see Fig. 4a and b). Set Tr := △C D B,

Tru := △C E B,

and Trl := △E D B .

78


The triangles Tr , Tru , Trl are independent of j. We will now show that ∥u 0 − v j ∥ L 2 (BC) ≤ C/j ,

(6.7)

∥u 0 − v j ∥ L 2 (D B) ≤ C/j ,

(6.8)

for j large and the constant C is independent of j. (a) We first consider the case 0 < ξ j ≤ η j < 21 . Since η j < 1/2, we have Tru ⊂ T22, j (see Fig. 4a) and therefore u 0 − v j ∈ P 2 (Tru ). Now using the trace inequality, Poincare inequality, the fact that (u 0 − v j )(B) = 0, and (6.6) we have [ ] ∥u 0 − v j ∥ L 2 (BC) ≤ C∥u 0 − v j ∥1,Tru ≤ C |(u 0 − v j )(B)| + |u 0 − v j |1,Tru = C|u 0 − v j |1,Tru ≤ C|u 0 − v j |1,T ≤ C/j , ∫ where the constant C is independent of j. Moreover, since from above we have | BC (u 0 − v j ) ds| is bounded, using (6.6) we get ⏐] ⏐∫ [ ⏐ ⏐ ∥u 0 − v j ∥1,Tr ≤ C |u 0 − v j |1,Tr + ⏐ (u 0 − v j ) ds ⏐ BC [ ] ≤ C |u 0 − v j |1,T + ∥u 0 − v j ∥ L 2 (BC) ≤ C/j , and consequently using trace inequality, ∥u 0 − v j ∥ L 2 (D B) ≤ C∥u 0 − v j ∥1,Tr ≤ C/j , where C is independent of j. (b) We now consider 21 < η j < 1. Now Trl ⊂ T21, j (see Fig. 4b) and u 0 − v j ∈ P 2 (Trl ). Since (u 0 − v j )(B) = 0, using the same arguments, we get ∥u 0 − v j ∥ L 2 (D B) ≤ C∥u 0 − v j ∥1,Trl ≤ C|u 0 − v j |1,Trl ≤ C|u 0 − v j |1,T ≤ C/j . Also ∥u 0 − v j ∥1,Tr ≤ C[|u 0 − v j |1,Tr + ∥u 0 − v j ∥ L 2 (D B) ] ≤ C/j , and therefore, ∥u 0 − v j ∥ L 2 (BC) ≤ ∥u 0 − v j ∥1,Tr ≤ C/j . The proof for η = 1/2 is similar. Thus (6.7) and (6.8) hold for all 0 < ξ ≤ η < 1. Since v j ∈ S2 (ξ j , η j ), by definition v j | BC = 0 for all j. Therefore from (6.7), we have ∥u 0 ∥ L 2 (BC) = ∥u 0 − v j ∥ L 2 (BC) ≤ C/j,

∀ j,

and since u 0 is independent of j, we have u 0 | BC = 0. Recall that u 0 − v j is a quadratic on M j B and |D B| < |M j B|. Therefore, using the equivalence of norms and (6.8), |u 0 (M j ) − v j (M j )| ≤ ∥u 0 − v j ∥ L ∞ (M j B) ≤ C∥u 0 − v j ∥ L ∞ (D B) ≤ C∥u 0 − v j ∥ L 2 (D B) ≤ C/j ,

(6.9)

|M j B| |D B|

< 2; thus C is independent of j. for j large enough and where C depends on Now consider the point N ∗j on AC such that |AN ∗j | = |AM j | = ξ j ≤ η j (see Fig. 4c) and set Tl, j := △N ∗j AM j ⊂ T1, j . Clearly Tl, j is equilateral and we have w j := u 0 − v j ∈ P 2 (Tl, j ). For each j, we map Tl, j to⏐ the master ⏐triangle T¯ = △ A¯ B¯ C¯ with A¯ ↔ A, B¯ ↔ M j , and C¯ ↔ N ∗j . Then by standard augments, w j ⏐T ↔ w¯ j ⏐T¯ with l, j ¯ = w j (A), w¯ j ( B) ¯ = w j (M j ), and w¯ j (C) ¯ = w j (N ∗j ). w¯ j = u¯ 0 − v¯ j ∈ P 2 (T¯ ), w¯ j ( A)


79

¯ Q j ⇔ Q; ¯ F¯ is the mid-point of A¯ C. ¯ Fig. 5. (a) |P j∗ B| = |M j B|; (b) △P j∗ B M j ⇔ △ A¯ B¯ C;

⏐ Note that w¯ j ⏐ A¯ B¯ is quadratic on the side A¯ B¯ of T¯ . We use the equivalence of norms and the Trace inequality on T¯ to get ¯ ≤ ∥w¯ j ∥ L ∞ ( A¯ B) |w¯ j ( A)| ¯ j ∥ L 2 ( A¯ B) ¯ ≤ C∥w ¯ [ ¯ ≤ C∥w¯ j ∥1,T¯ ≤ C |w¯ j ( B)| + |w¯ j |

1,T¯

]

,

and therefore mapping back to Tl, j and using (6.6), (6.9), we have [ ] |u 0 (A) − v j (A)| ≤ C |u 0 (M j ) − v j (M j )| + |u 0 − v j |1,Tl, j ≤ C/j,

for j large enough,

where C is independent of j. Since v j ∈ S2 (ξ j , η j ), we have v j (A) = 0 for all j. Therefore, from above we have |u 0 (A)| ≤ C/j

for j large enough,

⏐ and since u 0 is independent of j, we conclude that u 0 (A) = 0. We have shown before that u 0 ⏐ BC = 0, which implies that u 0 ≡ 0, which contradicts the fact that |u 0 |1,T = 1. Thus (6.4) holds for all 0 < ξ ≤ 1/2 and ξ ≤ η ≤ 1. Case 2: We now consider 1/2 ≤ ξ < 1, and suppose (6.4) does not hold. Then following the same arguments as before, there exist sequences {ξ j }, {η j }, {u j ∈ S1∗ }, {v j ∈ S2 (ξ j , η j )}, u 0 ∈ S1∗ where 1/2 ≤ ξ j < 1, ξ j ≤ η j ≤ 1, and (6.5), (6.6) hold. Consider the point P j∗ on BC such that |P j∗ B| = |M j B| = 1 − ξ j > 0 (see Fig. 5a). Let P j∗ M intersect N j B at Q j . Set Tr, j := △P j∗ B M j , Tr,u j := △P ∗ Q j B, Tr,l j := △Q j M j B . Note Tr, j ⊂ T21, j ∪ T22, j , Tr,u j ⊂ T22, j and Tr,l j ⊂ T21, j . Therefore, setting w j := u 0 − v j we have ⏐ ⏐ w j ⏐T u ∈ P 2 (Tr,u j ), w j ⏐T l ∈ P 2 (Tr,l j ), and w j (B) = 0 . r, j

r, j

For each j, we map equilateral triangle Tr, j to the master equilateral triangle T¯ := △ A¯ B¯ C¯ such that M j ↔ ¯ ¯ Let Q j ↔ Q¯ and F¯ be the mid-point of the side A¯ C¯ of the master triangle A¯ B¯ C¯ (see A, B ↔ B¯ and P j∗ ↔ C. Fig. 5b). We set ¯ Tr,u j ↔ T¯ u := △C¯ Q¯ B, T¯L := △ F¯ A¯ B¯ . ⏐ ⏐ ¯ = 0. Observing that Then by standard arguments we have w j ↔ w, ¯ w¯ ⏐T¯ l ∈ P 2 (T¯ l ), w¯ ⏐T¯ u ∈ P 2 (T¯ u ), and w( ¯ B) T¯L ⊂ T¯ l we have [ ] ¯ + |w| ∥w∥ ¯ L 2 ( A¯ B) ¯ 1,T¯L ≤ C |w( ¯ B)| ¯ 1,T¯L = C|w| ¯ 1,T¯L ≤ C|w| ¯ 1,T¯ . ¯ ≤ C∥w∥ ⏐ ⏐ Next since v¯ j ⏐ B¯ C¯ = 0, we have w¯ ⏐ B¯ C¯ = u¯ 0 is linear. Therefore [ ] ¯ ≤ ∥w∥ |w( ¯ C)| ¯ L ∞ ( B¯ C) ¯ L 2 ( B¯ C) ¯ 1,T¯ ≤ C |w| ¯ 1,T¯ + |w| ¯ L 2 ( A¯ B) ≤ C|w| ¯ 1,T¯ . ¯ ≤ C∥w∥ ¯ ≤ C∥w∥ ¯ ¯ Tr,l j ↔ T¯ l := △ Q¯ A¯ B,

80


Therefore, mapping back to Tr, j and using (6.6) we get |u 0 (P j∗ ) − v j (P j∗ )| ≤ C|u 0 − v j |1,Tr, j ≤ C/j . ⏐ of j, we have u 0 (P j∗ ) = 0 from above. Moreover, since u 0 ⏐ BC ∈ P 1 (BC) Since v j (P j∗ ) = 0, and u 0 is independent ⏐ and u 0 (B) = 0, we infer that u 0 ⏐ BC = 0. Let F, D be the mid-points of AC, AB, respectively (see Fig. 5a). Set Tl∗ := △AF D ⊂ T1, j , ⏐ ⏐ where T1, j = △AM j N j as defined before. Since (u 0 − v j )⏐T ∈ P 2 (T1, j ), we have (u 0 − v j )⏐T ∗ ∈ P 2 (Tl∗ ). Now 1, j l ⏐ using the equivalence of norms on T ∗ , the fact (u 0 − v j )⏐ = 0, and (6.6), we get l

BC

|u 0 (A)| = |u 0 (A) − v j (A)| ≤ ∥u 0 − v j ∥ L ∞ (Tl∗ ) ≤ C∥u 0 − v j ∥ L 2 (Tl∗ ) ≤ C∥u 0 − v j ∥1,T ≤ C|u 0 − v j |1,T ≤ C/j . ⏐ Since u 0 is independent of j, it is clear from above that u 0 (A) = 0. We have shown before that u 0 ⏐ BC = 0, and since u 0 ∈ P 1 (T ), we have u 0 ≡ 0 on T which contradicts the fact that |u 0 |1,T = 0. Therefore (6.4) holds for all 1/2 ≤ ξ < 1 and ξ ≤ η ≤ 1 and thus it holds for all 0 < ξ < 1, ξ ≤ η ≤ 1. The result (6.4) also holds for 0 < η ≤ ξ < 1. It could be obtained by using same ideas in the proof of the result for the case 0 < ξ ≤ η < 1 and exploiting the symmetry about the perpendicular bisector of the master element T , passing through the vertex A; we do not elaborate it here. ■ Remark 6.1. We note that the line segment M N could intersect the sides AB, BC or AC, BC of T , as mentioned in the beginning of this section. By using the rotational symmetry and symmetry about the perpendicular bisectors of the equilateral master element T , it can be shown that Lemma 6.1 holds for all positions of the line segment M N in T . Thus the constant C in Lemma 6.1 is independent of the position of the line segment M N in T . Theorem 6.2. The spaces S1 and S2 satisfy the Axiom A1, i.e., [ ] [ ] L 1 ∥v1 ∥2E + ∥v2 ∥2E ≤ B(v1 + v2 , v1 + v2 ) ≤ U1 ∥v1 ∥2E + ∥v2 ∥2E for all v1 ∈ S1 and v2 ∈ S2 , where the constant L 1 , U1 are independent of h as well as the position of the interface Γ. Proof. Using Lemma 6.1, Remark 6.1, and the standard finite-element scaling argument, mapping an element τs with the master element, it is easy to show that (5.8) holds for all τs , s ∈ E h,enr and thus from Lemma 5.2 we have the desired result. ■ 7. Axiom A2 — Strong stability In this section, we will show that the enrichment space S2 satisfies the “element level” sufficient conditions (5.13) and (5.14) such that Axiom A2 holds. Especially, we will show that the constants C1 , C2 in (5.13) and (5.14) respectively are independent of the position of the interface in an element. Following the notations in Section 6 and referring to the master element T = △ABC in Fig. 3, we write any w ∈ S2T in the form w = aΦ A + bΦ B + cΦC ,

(7.1)

where Φi , i = A, B, C are locally scaled shape functions on T , namely Φi = ζi Ni ϕξ,η ,

ζi ∈ R are chosen such that |Φi |1,T = 1,

i = A, B, C.

(7.2)

We first state an important technical result. Lemma 7.1. Let w = aΦ A + bΦ B + cΦC ∈ S2T . Then there exists C > 0, independent of a, b, c, ξ and η, such that |w|21,T ≥ C(a 2 + b2 + c2 ) .

(7.3)

81


We will establish this result later in the section. Theorem 7.2. The space S2 satisfies Axiom A2, i.e., −1 2 2 T L 2 ∥D−1 22 x∥ ≤ x A22 x ≤ U2 ∥D22 x∥ ,

for all x ∈ R|Ih,enr |

where the constants L 2 , U2 are independent of h as well as the position of the interface Γh with respect to the mesh. Proof. We consider an enriched node xi , i ∈ Ih,enr . Recall that (see (5.3)) the scaled shape function Ψi of S2 , associated with xi , has compact support ωi and satisfies Bh (Ψi , Ψi ) = 1. Since Ψi |τs ≡ 0 for s ̸∈ E hi ∩ E h,enr , i.e., if τs is not an enriched element, we have ∑ ∑ Bh (Ψi , Ψi ) = Bh,τs (Ψi , Ψi ) = Bh,τs (Ψi , Ψi ) = 1. (7.4) s∈E hi

s∈E hi ∩E h,enr

Let τs ∗ ⊂ ωi be the element such that Bh,τs ∗ (Ψi , Ψi ) =

max

s∈E hi ∩E h,enr

Bh,τs (Ψi , Ψi ) .

Then τ˚s ∗ ∩ Γ ̸= ∅ and it is easy to see from (7.4) that C

−1

≤ ∥Ψi ∥2Eh ,τs ∗ = Bh,τs ∗ (Ψi , Ψi ) ≤ 1,

(7.5)

where C = maxi∈Ih,enr |E hi ∩ E h,enr | (recall |E hi ∩ E h,enr | = card{E hi ∩ E h,enr }); C is independent of i. Since τs ∗ ⊂ ωi is an enriched element, it is clear that xi is a vertex of τs ∗ . Let xl , xk be the other two vertices of τs ∗ , ∗ i.e., Ihs = {i, l, k}. For any δ = {δi , δl , δk } ∈ R3 , we consider the function v = δi Ψi + δl Ψl + δk Ψk

on τs ∗ ,

where Ψl , Ψk are the scaled (not locally scaled) shape function of S2 associated with the nodes xl , xk , respectively, satisfying Bh (Ψl , Ψl ) = Bh (Ψk , Ψk ) = 1. Clearly v = v| ˜ τs ∗ for some v˜ ∈ S2 and ∗

(s ) ∥v∥2Eh ,τs ∗ = δ T A22 δ

(7.6)

∗

) ∗ where A(s 22 = {Bh,τs ∗ (Ψm , Ψn )}m,n∈Ihs (see (5.5)). We next consider the affine mapping Fs ∗ : T → τs ∗ where T is the master equilateral triangle. Then v is mapped to v¯ ∈ S2T given by

v¯ = δi Φ¯ i + δl Φ¯ l + δk Φ¯ k , where Φ¯ m = Fs−1 ∗ (Ψm |τs ∗ ),

∗

m ∈ Ihs = {i, k, l} .

Now using (7.5), (2.3), and a standard scaling argument, we have D2 D1 2 C1 ≤ ∥Ψi ∥2Eh ,τs ∗ ≤ D1 |Ψi |21,τs ∗ ≤ |Φ¯ i |1,T ≤ D2 |Ψi |21,τs ∗ ≤ ∥Ψi ∥2Eh ,τs ∗ ≤ C 2 , β1 β0 where the positive constants D1 , D2 depend on the “minimal angle condition” of the mesh and C 1 = D1 C D2 /β0 . Thus the constants C 1 , C 2 are independent of s ∗ and i. We set ˜m = ζ¯m Φ¯ m , Φ

(7.7) −1

/β1 , C 2 =

∗

m ∈ Ihs ,

˜m |1,T = 1. It is clear from (7.7) that where ζ¯m ∈ R is such that |Φ C2

−1/2

−1/2 ≤ |ζ¯i | ≤ C 1 .

Note that |ζ¯l |, |ζ¯k | may not be bounded from above or below. We rewrite v¯ as δi ˜ δl ˜ δk ˜ v¯ = Φ Φl + Φ i + k. ζ¯i ζ¯l ζ¯k

(7.8)

82


˜i |1,T = |Φ ˜l |1,T = |Φ ˜k |1,T = 1 as required in (7.2). Now from Lemma 7.1, Note that v¯ is of the form (7.1) with |Φ (2.3), (7.6), (7.8), and a standard scaling argument, we have ∗

1 2 |v| ≥ β1 1,τs ∗ δl2 δk2 ] DC [ δi2 ≥ + + ≥ β1 ζ¯i2 ζ¯l2 ζ¯k2

(s ) δ T A22 δ = ∥v∥2Eh ,τs ∗ ≥

D 2 |v| ¯ β1 1,T DC δi2 DCC 1 2 ≥ δ , β1 ζ¯i2 β1 i

(7.9)

where the constant D depends on the minimal angle condition of the mesh and the constant C > 0 is from Lemma 7.1. Thus by setting C1 = DCC 1 /β1 in (7.9), we get (5.13) of Lemma 5.3. We note that the constant C1 > 0 is independent of i, h, and the position of Γh in τs ∗ . Next let τs , s ∈ E h,enr be an arbitrary enriched element. Let xa , xb , xc be the vertices of τs and suppose Ψa , Ψb , Ψc are the scaled shape function of S2 associated with the vertices xa , xb , xc , respectively. Then for any δ = (δa , δb , δc ) ∈ R3 , 2 δ T Aˆ (s) 22 δ = ∥w∥Eh ,τs ,

(7.10)

where w = δa Ψa + δb Ψb + δc Ψc . Now since ∥Ψi ∥2Eh = Bh (Ψi , Ψi ) = 1 for i ∈ Ih,enr , we have ∥Ψa ∥2Eh = ∥Ψb ∥2Eh = ∥Ψc ∥2Eh = 1, and therefore ∥w∥Eh ,τs ≤ |δa | ∥Ψa ∥Eh ,τs + |δb | ∥Ψb ∥Eh ,τs + |δc | ∥Ψc ∥Eh ,τs ≤ |δa | + |δb | + |δc | ≤ C2 ∥δ∥, √ where C2 = 3. Hence from (7.10) we have 2 δ T Aˆ (s) 22 δ ≤ C 2 ∥δ∥ ,

for all δ ∈ R3 ,

where the constant C2 is independent of i, h, and the position of Γh in τs , s ∈ E h,enr , which is precisely (5.14) in Lemma 5.3. Thus from (7.9) and Lemma 5.3, there exist constants L 2 , U2 such that −1 2 2 T L 2 ∥D−1 22 x∥ ≤ x A22 x ≤ U2 ∥D22 x∥ ,

for all x ∈ R|Ih,enr | ,

which is the desired result. ■ We will now establish (7.3) of Lemma 7.1. We will use the notations used in Section 6 and will refer to Fig. 3 where M N intersects the sides AB, AC of the master triangle T = △ABC. Proof of Lemma 7.1 when Γ is a straight interface. We note that the enrichment function ϕξ,η is linear on T2 := T21 ∪ T22 when the interface is straight. Indeed, ϕξ,η is linear on T1 and it is continuous, piecewise linear on T . We first consider 0 < ξ ≤ η < 1. Without loss of generality, we scale ϕξ,η such that ϕξ,η (M) = 1; it only changes the values of ζi in (7.2). Since ϕξ,η (N ) = 0 when N = C, ϕξ,η (N ) = ϕξ,η (M) = 1 when ξ = η, and ϕξ,η | N C is linear, we have 1−η . W := ϕξ,η (N ) = 1−ξ Moreover, it is clear from the definition of ζi that ζi = |Ni ϕξ,η |−1 1,T . A simple calculation shows that ⎧ [ ] 4 1 W2 W ⎪ ⎪ ⎪ + 2 − on T1 ; ⎨ 2 3 ξ η ξη 2 |∇ϕξ,η | = [ ] ⎪ 4 1 W2 W ⎪ ⎪ + 2 − on T2 . ⎩ 3 ξ2 η (1 − ξ )(1 − η)

(7.11)

For the clarity of notation, we use S2 instead of S2T . Let M L < M R be the points on the edge AB of T such that |AM L | = ξ0l , |AM R | = ξ0r , with 0 < ξ0l < 1/2 < ξ0r < 1; ξ0l , ξ0r are fixed. Case 1: We first consider ξ0l ≤ ξ ≤ ξ0r .


83

We first show that ζi , i = A, B, C are uniformly bounded from above and below, where the bounds depend only on ξ0l , ξ0r . Since ϕξ,η is bounded on T and Ni are hat-functions associated with the nodes A, B, C, it is immediate that from (7.11) that |Ni ϕξ,η |1,T ≤ C2 ,

for i = A, B, C, ξ0l ,

(7.12)

ξ0r .

where C2 depends only on To show that |Ni ϕξ,η |1,T is bounded from below, we assume that it is not true. Then there exist sequences {ξ j }, {η j } such that ξ0l ≤ ξ j ≤ ξ0r and 1 > η j ≥ ξ j such that |Ni ϕξ j ,η j |1,T < 1/j .

(7.13)

Note that the interface intersects the sides AB and AC at M j and N j respectively with |AM j | = ξ j and |AN j | = η j (see Fig. 3). Furthermore, ϕξ j ,η j (M j ) = 1 and ϕξ j ,η j (N j ) = (1 − η j )/(1 − ξ j ). Since the sequences {ξ j }, {η j } are bounded, there exist subsequences, denoted again by {ξ j }, {η j }, such that ξ j → ξ ∗ and η j → η∗ with ξ0l ≤ ξ ∗ ≤ ξ0r and 1 > η∗ ≥ ξ ∗ ≥ ξ0l . Let M ∗ , N ∗ be points on AB, AC respectively, such that |AM ∗ | = ξ ∗ , |AN ∗ | = η∗ . Set T1∗ := △AM ∗ N ∗ and T2∗ := T \ T1∗ . Also let K ∗ be the mid-point of M ∗ N ∗ and set ϕ ∗ := ϕξ ∗ ,η∗ . Since M j → M ∗ , N j → N ∗ ,⏐ we have ϕ ∗⏐(M ∗ ) = 1, ϕ ∗ (N ∗ ) = (1 − η∗ )/(1 − ξ ∗ ), and from (7.13) we have |Ni ϕ ∗ |1,T = 0. Therefore Ni ϕ ∗ ⏐T ∗ and Ni ϕ ∗ ⏐T ∗ are constants, and since Ni ϕ ∗ (A) = Ni ϕ ∗ (B) = Ni ϕ ∗ (C) = 0, we 1 ⏐ 2 ⏐ have Ni ϕ ∗ ⏐T ∗ = Ni ϕ ∗ ⏐T ∗ = 0. But ϕ ∗ is continuous on T , linear on T1∗ , T2∗ with ϕ ∗ (M ∗ ) = 1, it is immediate that 1 ⏐ 2 ⏐ Ni ϕ ∗ ⏐T ∗ = Ni ϕ ∗ ⏐T ∗ = 0 for i = A, B is true, only if Ni = 0 for i = A, B, which is a contradiction. 1

2

l r ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Also since ⏐ ϕ (N )∗= ⏐ (1 − η )/(1 − ξ ) and ϕ (M ) = 1, we have |ϕ (K )| > C, where C depends only on ξ0 , ξ0 . ∗⏐ ⏐ Thus Ni ϕ T ∗ = Ni ϕ T ∗ = 0 for i = C implies NC = 0 which is a contradiction. Thus 1

2

|Ni ϕξ,η |1,T ≥ C1 ,

for i = A, B, C,

(7.14)

where C1 depends only on ξ0l , ξ0r . Therefore from (7.12), (7.14), we have −1 C2−1 ≤ ζi = |Ni ϕξ,η |−1 1,T ≤ C 1 ,

i = A, B, C,

where C1 , C2 are independent of ξ, η, but may depend on ξ0l , ξ0r . We now prove (7.3), where without loss of generality, we assume that ζi = 1, Φi = Ni ϕξ,η for i = A, B, C and w = (a Ni + bN B + cNC )ϕξ,η . Suppose (7.3) is not true. Then there exist sequences {a j }, {b j }, {c j }, {ξ j }, {η j }, {M j }, {N j }, {w j } with ξ0l ≤ ξ j ≤ ξ0r and 1 > η j ≥ ξ j , such that a 2j + b2j + c2j = 1,

(7.15)

and |w j |1,T ≤ 1/j ,

(7.16)

where w j = L j ϕξ j η j and L j = a j N A + b j N B + c j NC . Since the sequences {a j }, {b j }, {c j }, {ξ j }, {η j } are bounded, there are subsequences, denoted again by {a j }, {b j }, {c j }, {ξ j }, {η j } with η j ≥ ξ j such that a j → a ∗ , b j → b∗ , c j → c∗ , ξ j → ξ ∗ , η j → η∗ satisfying ξ0l ≤ ξ ∗ ≤ ξ0r , 1 > η∗ ≥ ξ ∗ ≥ ξ0l and from (7.15), we have a ∗ 2 + b∗ 2 + c∗ 2 = 1.

(7.17)

We define M ∗ , N ∗ , K ∗ as before. Also from (7.16), we have |w ∗ |1,T = 0 where w∗ = L ∗ ϕ ∗ with ϕ ∗ := ϕξ ∗ ,η∗ and L ∗ = a ∗ N A + b∗ N B + c∗ NC . ⏐ ⏐ Let T1∗ , T2∗ be as defined before. Now |w ∗ |1,T = 0 implies that w∗ ⏐T ∗ and w∗ ⏐T ∗ are constants, and since w∗ (A) = ⏐ ⏐ 2 1 w∗ (B) = w ∗ (C) = 0, we have w ∗ ⏐T ∗ = w∗ ⏐T ∗ = 0. As before, since ϕ ∗ (M ∗ ) = 1, ϕ ∗ (A) = ϕ ∗ (B) = 0, we have 1

2

84


⏐ ⏐ ⏐ ∗ w∗ ⏐ AB = L ∗ ϕ ∗ ⏐ AB = 0 only if L ∗ ⏐ AB = 0, i.e., a ∗ = b∗ = 0. Again, on T , linear on T⏐ 1∗ , T2∗ , and ⏐ ϕ is continuous ⏐ l r ∗ ∗ ∗⏐ ∗ ∗⏐ |ϕ (K )| ≥ C > 0, where C depends only on ξ0 , ξ0 . Therefore w M ∗ N ∗ = L ϕ M ∗ N ∗ = 0 only if L ∗ ⏐ M ∗ N ∗ = 0, i.e, c∗ = 0. We thus have a ∗ = b∗ = c∗ = 0, which contradicts (7.17). Thus (7.3) is true with the constant C > 0 independent of ξ, η, but may depend on ξ0l , ξ0r . Case 2: We now consider 0 < ξ < ξ0l . We again refer to Fig. 3 and let w ∈ S2T be of the form (7.1) with ζi , i = A, B, C as defined in (7.2). It has been shown in Appendix that there exists constant C > 0 such that ζA2 ≥ C

ξ and ζ B 2 ≥ C, η

(7.18)

where C in independent of ξ, η, but may depend on ξ0l . We will prove (7.3) by contradiction. Suppose there exist sequences {a j }, {b j }, {c j }, {ξ j }, {η j }, {w j } with 0 < ξ j < ξ0l and η j ≥ ξ j such that a 2j + b2j + c2j = 1

(7.19)

and |w j |1,T
0 is independent of j, ξ j , η j , but depends on ξ0l , ξ0r . Now using (7.23) we get |b j | → 0

as j → ∞.

Next we set T1, j = △AM j N j and for (x, y) ∈ T1, j , we consider the mapping ηj ) 1 ( x +√ 1− y; x¯ = ξj ξj 3 ηj y y¯ = . ηj Clearly, T1, j is mapped to the equilateral triangle T ∗ := A∗ M ∗ N ∗ with A∗ = (0, 0), B ∗ = (0, 1), C ∗ = ( 12 ,

√ 3 ). 2


Now using change of variables, we have ∫ [( ∂w j )2 ( ∂w j )2 ] + |w j |21,T ≥ d x dy ∂x ∂y T1, j ∫ ∫ ( ∂w j )2 1 ( ∂ w¯ j )2 d x dy = ξ j η j d x¯ d y¯ ≥ 2 ∂x ∂ x¯ T∗ ξj T1, j ∫ ( ηj ∂ w¯ j )2 = d x¯ d y¯ . ξ j T ∗ ∂ x¯

85

(7.24)

j j From (7.21), it is clear that ℓ j (A) = ℓ j (0, 0) = a j ζ A , therefore ℓ¯ j (A∗ ) = a j ζ A , where A∗ is the vertex of T ∗ associated with the vertex A of T1, j . Moreover, since ϕξ j ,η j is linear on T1, j , ϕξ j ,η j (A) = 0, ϕξ j ,η j (M j ) = 1, we ∂ ϕ¯ξ ,η

have ϕ¯ξ j ,η j is linear on T ∗ , ϕ¯ξ j ,η j (A∗ ) = 0, ϕ¯ξ j ,η j (B ∗ ) = 1 and ∂jx¯ j = 1 on T ∗ . Therefore it is easy to check that j ∂ w¯ (A∗ ) = ∂∂wx¯¯ (0, 0) = a j ζ A and thus, since ∂∂wx¯¯ is linear on T ∗ , we have ∂ x¯ ∂ w¯ j j = a j ζ A + Pξ j ,η j x¯ + Q ξ j ,η j y¯ , ∂ x¯ where Pξ j ,η j , Q ξ j ,η j are constants depending on ξ j , η j . Therefore from (7.24), (7.20), (7.18), and using the equivalence of norms of linear polynomials on T ∗ , there exists a constant C > 0, independent of ξ j , η j but may depend on ξ0l , such that ∫ ηj 1 j 2 > |w j |1,T ≥ (a j ζ A + Pξ j ,η j x¯ + Q ξ j ,η j y¯ )2 d x¯ d y¯ 2 ∗ j ξ j T1 ηj 2 j2 [a ζ + Pξ2j ,η j + Q 2ξ j ,η j ] ≥ Ca 2j , ≥ ξj j A which implies that a j → 0,

for j → ∞.

Now since a j → 0, b j → 0 as j → ∞, it is clear from (7.19) that c j → 1 as j → ∞. Therefore using the fact j that |Φi |1,T = 1 for i = A, B, C, we have j

j

j

|w j |1,T = |a j Φ A + b j Φ B + c j ΦC |1,T ⏐ ⏐ 1 ≥ ⏐ |c j | − (|a j | + |b j |) ⏐ ≥ , for j large enough, 2 which contradicts (7.20). Thus (7.3) holds for 0 < ξ < ξ0l and η ≥ ξ . Using similar arguments, we can obtain (7.3) for ξ0r ≤ ξ < 1 and η ≥ ξ . Thus we have proved Lemma 7.1 for 0 < ξ ≤ η < 1, 0 < ξ < 1. Lemma 7.1 also holds for 0 < η ≤ ξ < 1. It could be obtained by using same ideas in the proof of the result for the case < ξ ≤ η < 1 and exploiting the symmetry about the perpendicular bisector of the master element T passing through the vertex A; we do not elaborate it here. ■ We considered the situation when M N intersects the master triangle △ABC. However, M N could intersect AB, BC or AC, BC depending on the position of Γ (straight interface) inside an element. As mentioned in Remark 6.1, it could be shown that Lemma 7.1 also holds for such situations by using the rotational symmetry and symmetry about the perpendicular bisectors on the equilateral master triangle T . Computational verification of Lemma 7.1 when Γ is a closed curved interface: Instead of an equilateral master triangle, we consider the right angled master triangle T = △ABC (as shown in Fig. 6) with the vertices A = (0, 0), B = (1, 0), and C = (0, 1). For the parameters 0 < ξ ≤ η < 1, we set the points M = (ξ, 0) and N = (0, η) on the sides AB and AC, respectively. Recall that M N is mapped to Γh |τs where τs = Fs (T ) is an enriched element. Also recall that the enrichment function ϕξ,η , in the case of the curved interface Γ , is continuous and piecewise linear on T . For ξ ≤ η, it is linear on T1 := △AN M, T21 := △N M B, and T22 := △C N B (refer to (3.8) in Section 3) (for ξ ≥ η, the triangles T21 , T22 are defined using the diagonal MC to maintain the maximal angle condition). Note that the enrichment function is different from that for the straight interface where it was linear on the quadrilateral T21 ∪ T22 .

86


Fig. 6. Geometry with the parameters ξ = 3/7, η = 4/7, xd = 0.9, and yl = 2.5.

We note that from (7.1), with T as defined above, we have ) |w|21,T = aT Aˆ (T 22 a,

a = (a, b, c) ∈ R3

(7.25)

where ) Aˆ (T 22 :=

{∫

} ∇Φk · ∇Φl d x T

l,k∈{A,B,C}

) and Φi , i = A, B, C, are as in (7.2) with respect to the right angled master triangle T . Clearly Aˆ (T 22 depends on ξ , η, ) and ϕξ,η . We will establish (7.3) of Lemma 7.1 by showing computationally that the smallest eigenvalue λm ( Aˆ (T 22 ) of (T ) ˆ A22 is bounded away from 0, for 0 < ξ ≤ η < 1. The values of the enrichment function ϕξ,η on T depend on the distance of the vertices of the enriched element τs = Fs (T ) from the interface Γh . It is important to note that the points on Γh , associated with distance from the ) vertices of τs , may not be on Γh |τs ; they could be on Γh inside the elements adjacent to τs . Therefore Aˆ (T 22 depends also on the position of Γh in the elements adjacent to the element τs . Since we have assumed that the curvature of Γ is bounded (small), it is clear that Γh in the elements adjacent to τs do not “bend too much away” from Γh |τs . We mimic this situation in the master triangle T by considering two points

P = (−1, yl ) on the vertical line x = −1, and Q = (xd , −1) on the horizontal line y = −1, with yl and xd as parameters. The angle between the line segments P N , N M and N M, M Q (see Fig. 6), which depends on the parameters yl and xd , mimics the “bend” in Γh from the element τs to its adjacent elements. Thus the ) ˆ (T ) minimum eigenvalue λm ( Aˆ (T 22 ) of A22 depends on the 4 parameters ξ, η, yl , x d . Our goal is to minimize the smallest (T ) eigenvalue λm ( Aˆ 22 ) with respect to ξ, η, yl , xd within their respective ranges. In order to control the angle between the line segments P N , N M and N M, M Q (i.e., the curvature of the interface), we include the following constraints on the parameters (ξ, η, xd , yl ): 10−12 ≤ ξ ≤ η ≤ 1 − 10−12 ,

xd ≥ xmin ,

yl ≥ ymin ,

where xmin = ξ +

1 − (η/ξ ) tan(π/12) , η/ξ + tan(π/12)

ymin = η +

η/ξ − tan(π/12) . 1 + (η/ξ ) tan(π/12)

Note that xmin , ymin depend on the values of ξ, η. We also include upper bounds on xd , yl which depend on how close ξ, η are to 1 respectively: if ξ ≤ 1 − 10−6 , then xd ≤ xmax , else xd ≤ xd⋆ ,


87

where xmax = ξ +

1 + (η/ξ ) tan(π/12) , η/ξ − tan(π/12)

xd⋆ = ξ +

ξ . η

Similarly if η ≤ 1 − 10−6 , then yl ≤ ymax , else yl ≤ yl⋆ , where η η/ξ + tan(π/12) , yl⋆ = η + . 1 − (η/ξ ) tan(π/12) ξ The upper and lower bounds on xd , yl are designed to ensure that the line segments P N , M Q do not bend too much at M, N : we only allow a deviation of ±π/12 from the “extended interface MN”. Furthermore, the upper bounds xd⋆ , yl⋆ are designed so that P N , M Q do not re-enter element T . The constrained minimization procedure is carried out in MATLAB using the fmincon function,which handles inequality constraints on the variables. Several choices for the minimization algorithm are available in fmincon, namely the interior-point, SQP (Sequential Quadratic Programming), and active set methods. All the three methods were used to find the minimum of λm by using the “built-in” stopping criteria of fmincon, namely, (a) “maximum number of function evaluations allowed: 3000”, (b) maximum number if iterations allowed: 400”, (c) “termination tolerance on the first order optimality: 10−6 ”, (d) tolerance on constraint violation: 10−6 ”. Random initialization choices in the admissible set were used to start the optimization process. For all the initial choices, we obtained ymax = η +

) min λm ( Aˆ (T 22 ) = 0.37844397.

(7.26)

ξ,η,xd ,yl

The critical values of the parameters, where the minimum is attained, are ξ = 0.72448079, η = 1–10−12 , xd = 1.1264189, and yl = 2.1032686. fmincon reported successful completion with the message “The relative first order optimality measure 4.7e−08 is less than options.Optimality Tolerance = 1.0e−06 and relative maximum ) constraint violation 0.0e+00 is less than options.Constraint tolerance = 1.0e−06”. Since the value of min λm ( Aˆ (T 22 ) in −8 (7.26) is only “accurate” up to 4.7 × 10 , we safely conclude that ) min λm ( Aˆ (T 22 ) > 0.3

(7.27)

ξ,η,xd ,yl

The procedure fmincon showed several local minima of the constrained minimization problem. However, we note that the same minimum value of min λm was attained at these local minima. We present some of the values of the parameters where min λm = 0.37844397 is attained. ξ

η

xd

yl

0.72448079 0.72448079 0.72448079 0.72448079 0.72448079 0.72448079

1–10−12

1.1264189 1.1068189 1.1066566 1.1851587 1.2995625 1.2828294

2.1032686 2.1502686 2.022313 1.9406967 2.3152043 2.1164733

1–10−12 1–10−12 1–10−12 1–10−12 1–10−12

It appears from our computations that min λm is attained for the “unique” critical values of ξ = 0.72448079 and η = 1–10−12 , but the critical values of xd , yl are not unique. For “confidence” in the computed min λm = 0.37844397, we fixed 3 of the 4 critical values of the parameters, ξ = 0.72448079, η = 1–10−12 , xd = 1.1264189, yl = 2.1032686 and plotted the values of λm against the free parameter in Fig. 7. It is clear from Fig. 7a, b that λm attains minimum at the critical values ξ = 0.72448079, η = 1–10−12 respectively. It appears from Fig. 7c, d that λm does not change with xd , yl respectively; the apparent constant value of λm is equal to min λm = 0.37844397. Now it is clear from (7.25) and (7.27) that [ ] ) 2 2 2 2 |w|21,T ≥ min λm ( Aˆ (T 22 ) ∥a∥ > 0.3(a + b + c ). ξ,η,xd ,yl

Finally mapping the right angled master element, given above, to the equilateral master element with side-lengths 1 (see (A.7)), we get (7.3) which is the desired result for 0 < ξ ≤ η < 1. The analogous result for the case 0 < η ≤ ξ < 1 could be obtained using the symmetry of the perpendicular bisector of the equilateral master element T passing through the vertex A. ■

88


Fig. 7. (a) Plot of λm against ξ for fixed η = 1–10−12 , xd = 1.1264189, yl = 2.1032686; (b) Plot of λm against η for fixed ξ = 0.72448079, xd = 1.1264189, yl = 2.1032686; (c) Plot of λm against xd for fixed ξ = 0.72448079, η = 1–10−12 , yl = 2.1032686; (d) Plot of λm against yl for fixed ξ = 0.72448079, η = 1–10−12 , xd = 1.1264189.

8. Conclusions In this paper, we presented a detailed and complete mathematical analysis of the GFEM when applied to smooth interface problems in 2D and showed that the GFEM is strongly stable. We have established the optimal order of convergence of the approximate solution. We have also shown that the conditioning of the GFEM, associated with the enrichment space presented in this paper (also in [32]), is not worse than that of the standard finite element method and the conditioning is robust with respect to the position of the mesh relative the interface, i.e., the GFEM is a strongly stable GFEM. We mention that though all the constants in various results in this paper are independent of the position of the mesh, they indeed depend on the contrast β1 /β0 of the interface problem. The mathematical framework presented in this paper could also be used for the interface problem in 3D, however one may have to use, at places, different mathematical details related to 3D. We finally mention that one has to use a different enrichment space for non-smooth interface problems, which will be addressed in a forthcoming paper. Appendix √

We consider the equilateral triangle T := △ABC with√vertices A = (0, 0), B = (1, 0), and C = ( 21 , 23 ). For given parameters 0 < ξ ≤ η < 1, let M = (ξ, 0) and N = ( η2 , 23η ) be points on the sides AB and AC respectively. Clearly |AM| = ξ , |AN | = η. We set T1 := △AM N and T2 := T \ T1 .


89

Let Φ A := ζ A N A ϕξ,η ,

Φ B := ζ B N B ϕξ,η ,

Φc := ζC NC ϕξ,η

(A.1)

be the scaled shape functions on T associated with the nodes A, B, C, respectively, where ◦ N A , N B , NC are the hat-functions associated with the vertices A, B, C of the △ABC ◦ The enrichment function ϕξ,η is a continuous, piecewise linear function on T such that ϕ(A) = ϕ(B) = ϕ(C) = 0, ϕ(M) = 1 and ϕξ,η is linear on T1 and T2 . ◦ ζ A , ζ B , ζC are positive constants such that |Φi |1,T = 1, i = A, B, C. The constants ζ A , ζ B , ζC depend on the parameters ξ, η. Proposition. Let 0 < ξl ≤ 12 be fixed and consider 0 < ξ ≤ ξl . Then there exist constants C1 > 0, C2 > 0, independent of ξ, η but may depend on ξl , such that ζ A2 ≥ C1

ξ η

and

ζ B2 ≥ C2 .

Proof. We first prove the result for the right angled triangle T¯ := △ A¯ B¯ C¯ with vertices A¯ = (0, 0), B¯ = (1, 0), and C¯ = (0, 1). For given parameters 0 < ξ ≤ η < 1, let M¯ = (ξ, 0) and N¯ = (0, η) be points on the sides A¯ B¯ and A¯ C¯ ¯ = ξ , | A¯ N¯ | = η. We set T¯1 := △ A¯ M¯ N¯ and T¯2 := T¯ \ T¯1 . respectively. Clearly | A¯ M| ¯ B, ¯ C¯ are given by The hat functions associated with the nodes A, N¯ A¯ (x, ¯ y¯ ) = 1 − x¯ − y¯ ,

N¯ B¯ (x, ¯ y¯ ) = x, ¯

N¯ C¯ (x, ¯ y¯ ) = y¯ .

(A.2)

The enrichment function ϕ¯ξ,η (x, ¯ y¯ ) be a continuous, piecewise linear function on T¯ such that ϕ¯ξ,η (0, 0) = ϕ¯ξ,η (1, 0) = ϕ¯ξ,η (0, 1) = 0, ϕ¯ξ,η (ξ, 0) = 1 ϕ¯ξ,η |T¯1 and ϕ¯ξ,η |T¯2 are linear functions. It is easy to check that ⎧ x¯ W ⎪ ⎪ ⎨ ξ + η y¯ ϕ¯ξ,η (x, ¯ y¯ ) = ⎪ 1 − x¯ − y¯ ⎪ ⎩ 1−ξ where W :=

1−η . 1−ξ

(x, ¯ y¯ ) ∈ T¯1 (A.3) (x, ¯ y¯ ) ∈ T¯2

¯ B, ¯ C, ¯ we define We note that ϕ¯ξ,η ( N¯ ) = ϕ¯ξ,η (0, η) = W . Associated with the vertices A,

Φ¯ A¯ := ζ¯ A¯ N¯ A¯ ϕ¯ξ,η ,

Φ¯ B¯ := ζ¯ B¯ N¯ B¯ ϕ¯ξ,η ,

Φ¯ C¯ := ζ¯C¯ N¯ C¯ ϕ¯ξ,η

where ζ¯ A¯ , ζ¯ B¯ , ζ¯C¯ are positive constants such that |Φ¯ A¯ |1,T = |Φ¯ B¯ |1,T = |Φ¯ C¯ |1,T = 1. ¯ B, ¯ C, ¯ we have We first note that since |Φ¯ i |1,T = 1 for i = A, ∫ 2 2 −2 ζ¯i = | N¯ i ϕ¯ξ,η |1,T¯ = |∇( N¯ i ϕ¯ξ,η )| d x¯ d y¯ T¯ ∫ ∫ 2 2 = |∇( N¯ i ϕ¯ξ,η )| d x¯ d y¯ + |∇( N¯ i ϕ¯ξ,η )| d x¯ d y¯ T¯1

(A.4)

T¯2

where ∂∂x¯ ( N¯ i ϕ¯ξ,η ), ∂∂y¯ ( N¯ i ϕ¯ξ,η ) are linear polynomials on T¯1 and T¯2 , which could written explicitly from the expressions ¯ B, ¯ C. ¯ for ϕ¯ξ,η and N¯ i given above, for i = A,

90

I. Babuška et al. / Comput. Methods Appl. Mech. Engrg. 327 (2017) 58–92 −2 ¯ From (A.4) we get We first consider the constant ζ¯ A¯ , associated with the vertex A. [ η 2 1 −2 (η − 4η + 6) + (8 − 24η + 16η2 − 4η3 ) ζ¯ A¯ = 12(1 − ξ )2 ξ ] 2ξ 4ξ 2 ξ3 2 3 2 + (3 − 12η + 9η − 2η ) − (1 − 4η + η ) + (1 − 4η) η η η

where we have performed the integration with MAPLE. Now it is easy to show that for 0 < ξ < ξl , there exist positive constants L 1 , L 2 , independent of ξ, η but depending on ξl , such that η L 1η + L 2ξ −2 ζ¯ A¯ ≤ L 1 + L 2 = ξ ξ and consequently, 2 ζ¯ A¯ ≥

ξ ξ ≥ C1 , L 1η + L 2ξ η

C1 :=

1 L1 + L2

(A.5)

where C1 is independent of ξ, η but may depend on ξl . −2 ¯ Again performing the integration in MAPLE, We next consider the constant ζ¯ B¯ , associated with the vertex B. we get from (A.4) that [ ] ξ3 1 2 ¯ζ B¯ −2 = 2 + ξ η − 2ξ η + (1 − 2η) . 12(1 − ξ )2 η Now it is easy to show that for 0 < ξ < ξl , there exist positive constants L 3 , independent of ξ, η but may depend on ξl , such that −2 ζ¯ B¯ ≤ L3

and consequently there exists a constant C2 =

1 L3

2 ζ¯ B¯ ≥ C2

such that (A.6)

where C2 is independent of ξ, η but may depend on ξl . √ We now consider the equilateral triangle T := △ABC with vertices A = (0, 0), B = (1, 0), and C = ( 12 , 23 ). For (x, y) ∈ T , we consider the mapping 1 x¯ = x − √ y 3 2 y¯ = √ y. 3

(A.7)

¯ where A¯ = (0, 0), B¯ = (1, 0), C¯ = It is clear the T = △ABC is mapped to the right angled triangle T¯ = △ A¯ B¯ C, √ η 3η (1, 0). Moreover, M = (ξ, 0) ⇔ M¯ = (ξ, 0) and N = ( 2 , 2 ) ⇔ N¯ = (0, η). We set T¯1 = △ A¯ M¯ N¯ and T¯2 = T¯ \ T¯1 . The function Φ A , is mapped to Φ¯ A¯ = ζ A N¯ A¯ ϕ¯ξ,η where ζ A is as in (A.1), N¯ A¯ is the hat-function associated with the vertex A¯ and ϕ¯ξ,η |T¯ = ϕξ,η |T . Set α A¯ = |Φ¯ A¯ |1,T¯ Recalling that |Φ A |1,T = 1, it can be easily shown that α A¯ is bounded from above and below. We now define Ã¯ := ζ¯ A¯ N¯ A¯ ϕ¯ξ,η , Φ

where ζ¯ A¯ = α −1 ζA. A¯

Ã¯ | ¯ = 1. Thus we conclude from (A.5) that there exists a constant C, independent of ξ, η but may It is clear that |Φ 1,T depend on ξl , such that ζ¯ A¯ 2 = α −2 ζ2 ≥ C A¯ A

ξ η


91

and since, we have α −2 is bounded from below, independent of ξ, η, we have A¯ ξ η where C1 is independent of ξ, η but may depend on ξl . Following exactly the same argument, we have ζ A2 ≥ C1

ζ B2 ≥ C2 where C2 is independent of ξ, η but may depend on ξl .

□

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