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M˙adaptation

Mathematical Models and Methods in Applied Sciences c World Scientific Publishing Company

M-adaptation in the mimetic finite difference method

VITALIY GYRYA Applied Mathematics and Plasma Physics Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] KONSTANTIN LIPNIKOV Applied Mathematics and Plasma Physics Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] GIANMARCO MANZINI Applied Mathematics and Plasma Physics Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA, [email protected] DANIIL SVYATSKIY Applied Mathematics and Plasma Physics Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx) The mimetic finite difference method produces a family of schemes with equivalent properties such as the stencil size, stability region, and convergence order. Each member of this family is defined by a set of parameters which can be chosen locally for every mesh element. The number of parameters depends on the geometry of a particular mesh element. M-adaptation is a new adaptation methodology that identifies a member of this family with additional (superior) properties compared to the other schemes in the family. We analyze the enforcement of the discrete maximum principles for the diffusion equation in the primal and dual forms, the reduction of numerical dispersion and anisotropy for the acoustic wave equation, and the optimization of the performance of multi-grid solvers. Keywords: mimetic discretization; unstructured polyhedral meshes; discrete maximum principles; numerical optimization

1. Introduction The mimetic finite difference (MFD) method belongs to a class of compatible discretization methods6 . The mimetic/compatible schemes reproduce important properties of physical and mathematical models in the discrete framework. These properties include fundamental identities of the tensor and vector calculus, conservation laws, solution symmetry, and discrete maximum principles. The MFD method80 was originally introduced to the scientific community as the supportoperator method (SOM)103 . Later, the name of the method was changed to reflect better its mimetic nature. The SOM was originally developed to solve the linear diffusion problem in mixed 1

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form on quadrilateral and hexahedral meshes65, 67, 87, 104, 105 . However, since definitions of the primary mimetic operators (discrete gradient, curl and divergence) do not impose any restrictions on the shape of mesh elements, various ideas have been developed to extend the construction of the derived mimetic operators (dual to the primary operators) to unstructured polygonal and polyhedral meshes75, 83, 89 . The modern MFD technology works on almost arbitrary unstructured polygonal and polyhedral meshes that are not totally crazy34−36 . During the last decade, the MFD method has also been successfully employed to solve various PDEs. The list of treated problems includes electromagnetism32, 64, 66, 79, 85 , continuum mechanics97 , gas dynamics39, 86 , linear diffusion in primal form15, 31 , convection-diffusion10, 42 , Stokes11, 12, 14, 92 , elasticity9 , Reissner-Mindlin free plate7, 16, 19 , eigenvalues40 , obstacle problems5 , and two-phase flows in porous media84 . A posteriori estimators for the mimetic discretizations have been developed in Ref. [8, 17, 4]. High-order mimetic discretizations were developed in Refs. [58, 18, 13]. The MFD method produces a rich family of numerical schemes with equivalent properties (same size of the stencil, similar stability requirements, and same asymptotic convergence order) that can be easily parametrized. This is one of the major differences between the MFD and other compatible discretization methods, with a few notable exceptions. For a linear diffusion problem, similar families were identified in the mixed finite volume48 and hybrid finite volume54 methods. Typically, inside a single family, all schemes use the same primary mimetic operator (e.g., divergence) and differ by the choice of the derived mimetic operator (e.g., gradient). The analysis of this family commenced with the development of the SOM for diffusion problems on triangular and quadrilateral meshes68, 93 . However, a complete characterization of the family of schemes for various PDEs on polygonal and polyhedral meshes became possible after a more deeper understanding of the structure of the derived mimetic operators, more precisely, the structure of the involved inner product matrices. A first attempt in this direction is found in Ref. [25]; also pertinent to this issue are the works in Refs. [100, 119]. However, in these works the design of practical schemes is mainly focused on simplicial and quadrilateral meshes in 2D and this is a huge difference with the approach proposed by the MFD method, which is suitable to handle a much higher mesh complexity. The construction of the mimetic family includes a natural way of parametrization. The parameters are locally introduced at the elemental level and this feature simplifies the analysis of the family of mimetic schemes. On the other hand, the number of parameters grows almost quadratically with the number of faces (or nodes) in the element, which sometimes require new numerical tools to analyze the family even in the case of hexahedral meshes. A natural question arises here: Are some parameter values better than others? To answer this question, we need to formulate precisely our optimization criteria. Examples of additional useful properties that can be ensured by an appropriate choice of the parameters include the monotonicity or the positivity of the discrete solution and a reduced numerical dispersion and numerical anisotropy. The mimetic adaptation, or m-adaptation, is a new adaptation strategy that extracts a scheme with such additional properties from the family of mimetic schemes. The main goal of this work is to review a few interesting cases where the m-adaptation was particularly successful. The first example of m-adaptation is the selection of a scheme that satisfies a discrete maximum (and minimum) principle (DMP). The maximum and minimum principles are fundamental

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properties of the solutions of second-order elliptic problems as stated by the Hopf’s lemma under the condition that the diffusion tensor is continuously differentiable63 . More precisely, a solution of an elliptic problem in an open domain either is constant on the domain or cannot have an internal minimum (resp., maximum) whenever the source term is non-negative (resp., non-positive). Ensuring discrete versions of these principles is of major importance in the development of new numerical methods. For example, in multiphase flow and transport problems in heterogeneous porous media with anisotropic diffusivity, the DMP ensures accurate dynamics of a contaminant plume101, 109 . A huge amount of publications on compatible discretization methods investigate necessary and sufficient conditions for the DMP on simplicial meshes30, 70, 73 . Typically these conditions are formulated as geometric restrictions on mutual orientation of a computational mesh and tensorial problem coefficients52, 101, 108, 122 . Such conditions have been widely investigated, for example, in the case of anisotropic diffusion tensors for the multi-point flux approximation (MPFA) method1, 2, 50 in the O-formulation51 and the L-formulation3 , and its variants such as the enriched multi-point flux approximation (EMPFA) method43 and the full pressure support (FPS) method49, 56 . As pointed out in Ref. [72, 37, 71], no linear consistent nine-point control volume scheme constructed on square meshes with a very anisotropic tensor (or on very distorted quadrangular cells with an isotropic tensor) can respect the maximum or minimum principle. Nonetheless, this limitation can be overcome by looking for nonlinear schemes21 . To relax mesh assumptions and construct unconditional monotone schemes, nonlinear finite volume methods (even for linear PDEs) have been introduced for the diffusion equation24, 76−78, 88, 106, 107 and the advectiondiffusion equation22, 23, 90, 91, 95 . In these methods a numerical solution is obtained by solving the nonlinear system of equations. For some problems efficient solution of this nonlinear system is a challenging task. In the finite element framework, a nonlinear stabilization of a discrete method was shown to be a useful tool leading to the DMP38, 74 . Finally, we mention approaches based on the minization of energy functional subject to monotonicity constraints94, 99 . Different formulations of a DMP are possible since they can be derived from different formulations of the continuous maximum principle. One formulation of a DMP requires that the inverse of the stiffness system is a nonnegative matrix121 , which can be readily satisfied if the stiffness matrix is an M-matrix20 . In fact, a great number of finite difference26−29 and finite element44−46 methods for second-order elliptic problems yield M-matrices, which guarantees the monotonicity of the numerical method. Thus, the M-matrix condition becomes the key mathematical principle for the development of an m-adaptation strategy 81, 82 . The second example of m-adaptation is the reduction of the numerical dispersion and numerical anisotropy in modeling the propagation of acoustic waves. For this application, the idea of parameterizing a family of schemes is not new. One example is, for instance, a weighted combination of the standard (with the weight α) and the rotated (with the weight 1 − α) finite difference discretizations of the Laplace operator69, 110 . The sole parameter α is selected to improve the accuracy of the numerical scheme for the acoustic wave equation in the frequency formulation by minimizing the numerical anisotropy. Another example of a parametrized family of schemes is the finite element discretization of the acoustic wave equation on a square mesh in which the mass and stiffness matrices are computed approximately using a quadrature rule57, 129 . In this case, the two parameters are the positions of the quadrature points for the mass and the stiffness matrices. In

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Ref. [129] the optimal values of these parameters are found to minimize the numerical dispersion in the limit of finely resolved waves. The family of mimetic schemes on a square mesh contains the above scheme as well as many of the classical discretization schemes59, 61 . The third example of m-adaptation is the optimization of performance of black-box multigrid solvers. The most common approach to solve large systems of algebraic equations arising from the discretization of elliptic PDEs is to use a Krylov type iterative process. Its performance hinges on the effectiveness of the preconditioner and the efficiency with which the preconditioning system can be solved. For elliptic problems, algebraic multigrid (AMG) methods are well established, and deliver optimal scaling and efficiency120 . Here, we consider the AMG solver from the HYPRE library62 . This multigrid solver shows excellent performance for M-matrices; unfortunately, the mimetic family not always contain such a scheme. Therefore, we employ a simple optimization strategy based on the enforcement of the diagonal dominance for a stiffness matrix. The numerical experiments indicates that this allows the multigrid to use more aggressive coarsening strategy which in turn leads to a faster solver. The paper is organized as follows. In Section 2 we describe two different mimetic discretizations of the diffusion equation and highlight common design principles in their derivations. In Sections 3 and 4 we show how to select members from these two mimetic families that satisfy a DMP. In Section 5 we show how to combine the analysis of the mimetic families for the stiffness and mass matrices to find a scheme that reduces significantly numerical dispersion and numerical anisotropy. In Section 6 we discuss how to select a member of the family that leads to a better performance of the multigrid solvers. Final conclusions are offered in Section 7. 2. MFD method for diffusion problems Here we derive two different mimetic discretizations of the diffusion problem on polyhedral meshes: (i) a vertex-based discretization of the primary formulation in Section 2.2 and (ii) a cell/face-based discretization of the dual formulation in Section 2.3. Although the two discretizations are different, they are based on a common design principle. 2.1. Polyhedral mesh Let Ωh be a conformal partition of the domain Ω into polyhedral cells (we shall also call them elements) such that the intersection of any two distinct cells P and P0 is either a few common faces f, a few common edges e, a few common vertices v, or empty. Let |P| denote the cell volume, |f| denote the face area, and |e| denote the edge length. For each face f, we introduce a unit normal vector nf , which is fixed once and for all, i.e. its orientation does not depend on the cell under consideration. Let xP and xf denote the barycenters of cell P and face f, respectively. We consider the mesh regularity assumptions given in Ref. [31]. Assumption 2.1. There exist two positive real numbers N s > 0 and ρs > 0 such that every mesh Ωh admits a sub-partition Th into shape-regular simplexes such that • every cell P ∈ Ωh admits a decomposition Th |P made of less than N s simplexes;

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vi+1 xP

fi

ni

fi

ni

fi−1

fi−1 vi vi−1

ni−1

ni−1

Fig. 1. Geometric notation and location of degrees of freedom restricted to a polygonal cell P for the primal (left picture) and the dual (right picture) formulations. The circles and vectors represent degrees of freedom for the pressure and velocity, respectively. The stars represent additional pressure degrees of freedom in the hybrid method.

• the shape-regularity of simplex T ∈ Th is defined as follows: the ratio between the radius rT of the inscribed circle and the diameter hT is bounded from below: rT ≥ ρs . (2.1) hT These assumptions slightly restrict the shape of the admissible cells to avoid some pathological situations as h → 0. Nonetheless, mesh Ωh may contain ugly-shaped cells including non-convex or degenerate cells, as it can be observed for moving meshes and adaptive mesh refinements. In two-dimensions, a conformal polygonal partition of Ω is defined similarly. The mesh regularity assumptions are also similar to those in Assumption 2.1. 2.2. Mimetic discretization for the primary formulation Consider the diffusion problem in the primary formulation for the scalar unknown u − div(K∇u) = g,

(2.2)

subject, for simplicity, to homogeneous boundary conditions on ∂Ω. We will present the construction of the vertex-based mimetic discretization in such a way to elucidate the similarities and the differences with the classical Galerkin finite element discretization. Both discretization methods use the same discrete representation of function u(x) – the set of its values uv at the mesh vertices (see the left picture in Fig. 1): uv = u(xv ).

(2.3)

We collect these degrees of freedom restricted to a cell P into a vector uP = {uv }v∈P ∈ RnP . Note that the number of vertices nP of cell P may vary from cell to cell. The first difference between the classical FE method and the MFD method is in the interpretation of the underlying discrete functions. In the linear Galerkin FE method the vector uP of the degrees of freedom defines uniquely a linear function uh ∈ SP . Moreover, the definition is explicit. The continuity of local functions uh across mesh faces follows immediately from their definition. In

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the MFD method, the functions uh are not defined explicitly, and, in general, may not be unique. But we can always assume that SP ⊂ H 1 (P) ∩ C 0 (P). To formalize the relationship between the discrete and continuum spaces, we introduce the interpolation operator IP : RnP → SP and the projection operator ΠP : SP → RnP , which are such that uP = ΠP (uh )

and uh = IP (uP ).

The difference between the FE method and the MFD method in the definition of the local approximation space SP leads to different constructions of the local stiffness matrices as described in Sections 2.2.1. Once these matrices are defined, both methods produce a global system of algebraic equations following the same assembling procedure: X X NP MP uP = |P|NP gP , (2.4) P∈Ωh

P∈Ωh

where NP is the conventional assembling matrix, and the components of vector gP are associated with the loading term g(x). In the FE method, the latters are given by projecting the loading term onto the chosen set of basis functions. In the MFD method, we set gP = IP (g)/nP . The discrete problem is closed by setting essential boundary conditions for the related components of uP . 2.2.1. The elemental stiffness matrix If IP is given explicitly, the matrix MP can be calculated using a quadrature rule: Z uTP MP vP ≈ K∇(IP (uP )) · ∇(IP (vP )) dV.

(2.5)

P

In the case of a constant tensor K = KP and simplex P, equation (2.5) is an exact relation and the equality sign holds instead of “≈”. It can be shown that in this case, the FE and MFD methods generate the same stiffness matrix. For a non-constant K = K(x) a sufficiently accurate quadrature rule must be applied to evaluate the integral in (2.5). Different quadratures will produce different matrices. The situation changes drastically when P is a general polygon or a polyhedron. The existing polygonal/polyhedral FE methods are still based on the definition of an interpolation operator IP in SP 55, 112−115 . This interpolation operator is required to exactly reproduce linear polynomials, to be positive, and to be a linear combination of nodal interpolants that satisfy the Lagrange interpolation property, i.e., they take value 1 at the assigned vertex of P and are zero at all other vertices125−127 . These properties do not define IP uniquely, and its definition is usually based on some explicit geometric construction of the Lagrange interpolants mentioned above116−118 . Therefore, the evaluation of the integral in (2.5) may require a quadrature rule even when the diffusion tensor K is constant98 . In the MFD method, since the approximation functions are not explicitly defined, one cannot use formula (2.5) or its quadrature-based approximation. Instead, the stiffness matrix is built using a direct formula stemming from an exactness property also called the consistency condition, which we state as follows: for any linear polynomial q and any function vh ∈ SP it holds that: Z T ΠP (vh ) MP ΠP (q) = KP ∇q · ∇vh dV. (2.6) P

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The vector-matrix-vector product in the left-hand side of (2.6) is required to give the exact value of the volume integral in the right-hand side of (2.6). This property ensures that the mimetic approximation is second-order accurate. The volume integral in (2.6) can be calculated exactly if we impose weak additional restrictions to space SP . More precisely, a function vh ∈ SP must be integrated exactly on the faces f of cell P using a quadrature rule with the quadrature points at the vertices of f: Z X X vh dS = ωf,v vv , where ωf,v > 0 ∀v ∈ ∂P and ωf,v = |f|. f

v∈f

v∈∂P

Under these additional restrictions, by applying the Gauss-Green formula and noting that KP ∇q is a constant vector, we have: Z Z Z KP ∇q · ∇vh dV = KP ∇q · ∇vh dV = KP ∇q · vh nP dS P

P

= KP ∇q ·

∂P

XZ f∈∂P

vh nP,f dS = KP ∇q ·

f

X

nP,f

f∈∂P

X

ωf,v vv ,

v∈f

where nP,f is the unit vector that is orthogonal to f and pointing out of P. Due to linearity of the integration, the result has the form of a linear functional of ΠP (vh ). This means that there exists a vector rP (q) ∈ Rn that depends only on the geometry of cell P, the linear function q (we stress this fact explicitly), and the tensor KP such that Z KP ∇q · ∇vh dV = (ΠP (vh ))T rP (q) ∀vh ∈ SP . P

Inserting this relation in (2.6) and observing that ΠP (vh ) is an arbitrary vector in Rn , we obtain MP ΠP (q) = rP (q)

∀q ∈ P1 (P),

(2.7)

where matrix MP is still unknown. Due to linearity of matrix-vector operations, this gives only (d + 1) linearly independent conditions, where d is the spatial dimension. To derive a more convenient representation for the matrix equations (2.7), we introduce two rectangular matrices NP and RP . Let q1 = 1 and qi = xi , i = 2, . . . , d + 1. Then, NP = ΠP (q1 ) . . . , ΠP (qd+1 ) and RP = rP (q1 ) . . . , rP (qd+1 ) . The equations (2.7) are reduced to a single matrix equation: M P N P = RP .

(2.8)

(0) MP

Let denote a particular solution to (2.8). Then, the complete family of mimetic stiffness matrices is given by the parametric relation (0)

(1)

MP = MP + MP (α1 , . . . , αk ),

(2.9) (1)

which is expressed through the k parameters α1 , . . . , αk . Matrix MP ensures the stability of the MFD method. The total number of parameters is k = (nP − d − 1)(nP − d − 2)/2 and grows quadratically with nP . For example, k = 0 for a triangle and a tetrahedron, k = 1 for a quadrilateral, k = 3 for a triangular prism, and k = 10 for a hexahedron.

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2.2.2. Explicit construction of the mimetic stiffness matrix Matrix equation (2.8) can be solved using the standards tools of the linear algebra. It is easy to verify that the particular solution is given by (0)

MP = RP (RTP NP )† RTP ,

(2.10)

where (RTP NP )† is the pseudoinverse of matrix RTP NP . This matrix has rank d. The columns of matrix NP are the projections of the polynomial basis {1, x, y} in two dimensions and {1, x, y, z} in three dimenstions. The columns of matrix RP are obtained from (2.7). For example, using a local enumeration of the vertices and the faces of polygon P (see Fig. 1), e.g., vi and fi for i = 1, . . . , nP , we have the explicit formulas:     T fn n + f1 nT 1 xTv1 0 P fnP f1     T T   1 xTv  0 f1 n + f2 n 2  f1 f2    1 b P . (2.11) bP  = 1, N   KP = 0, R and R = NP =  P ..    ..  . . 2. .. .  .    . T T  0 fn −1 n + fn n 1 xT vnP

P

fnP −1

P

fnP

b P and R b P are two suitably defined nP ×2 rectangular matrices. Similar Here 1 = (1, 1, . . . , 1)T and N formulas hold in three dimensions. Note that the consistency condition holds true when vh is a linear function. Using this and straightforward calculations, we can show the following result. Lemma 2.1. Let NP and RP be as defined above. Then, ! 0 0T 0 0T T RP N P = = . bT N b 0 |P| KP 0R P P

(2.12)

The pseudoinverse matrix used in formula (2.24) is, thus, given by ! † 0 0T 0 0T T RP NP = bT N b −1 = 0 K−1 /|P| , 0 (R P P P) (0)

and matrix MP is equivalently given by the formula: (0)

b P (R bT N b −1 R bT . MP = R P P P) † (0) The rank of matrix RTP NP is equal to d; hence, matrix MP has also rank d and we need to introduce a correct stability term to obtain an admissible stiffness matrix MP and a convergent mimetic scheme, see Section 2.4. 2.2.3. Convergence of the mimetic method We recall the convergence result

31

in the mesh-dependent norm defined by X |||˜ v|||2 = vPT MP vP , P∈Ωh

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˜ is the vector of all degrees of freedom. Let Π be the global projection operator. The where v following result is proved in Ref. [31]. Theorem 2.2. Let {Ωh }h be a sequence of shape-regular mesh partitions of Ω, conveniently labeled by the mesh size parameter h for h → 0. Futhermore, let u(x) ∈ H 2 (Ω) ∩ H01 (Ω) be the solution ˜ be the numerical of problem (2.2) with the Dirichlet boundary conditions on ∂Ω. Finally, let u solution provided by the mimetic scheme. Then, there exists a positive constant C independent of h such that |||˜ u − Π(u)||| ≤ Ch |u|H 2 (Ω) + ||g||L2 (Ω) . 2.3. Mimetic discretization for the dual formulation Consider the dual formulation of diffusion problem (2.2): u = −K∇p, div u = g,

(2.13) (2.14)

subject, for simplicity, to homogeneous Dirichlet boundary conditions on ∂Ω. Here, the scalar function p(x) is referred to as the “pressure” and the vector function u(x) is referred to as the “velocity”. In the discrete form, the pressure function p(x) is represented by the cell-averaged values Z 1 pP = p dV, (2.15) |P| P which are associated with the mesh cells P. The velocity function u(x) is represented by the normal fluxes Z 1 uf = u · nf dS, (2.16) |f| f which are uf associated with the mesh faces f. This selection of degrees of freedom is commonly used in the MFD method, the mixed finite volume (MFV) method, the hybrid finite volume (HFV) method, and the lowest-order Raviart-Thomas mixed finite element (MFE) method. Following the structure of Section 2.2, we compare the MFD method with the MFE method. In this subsection, let nP denote the number of faces of element P. We can formally introduce a new interpolation operator IP : RnP → SP and a new projection operator ΠP : SP → RnP such that uP = ΠP (uh )

and uh = IP (uP ).

In both methods, the projection operator is defined by (2.16). In the MFE method, the interpolation operator is the inverse of the projection operator and is constructed explicitly. In the MFD method, the interpolation operator is not constructed explicitly and, in general, may not be unique. Hereafter, let 1 = (1, . . . , 1)T be the generic vector with all components equal to one. Both methods can be written in a mixed-hydrid form as MP uP − BP pP + CP λP = 0, BP uP = |P|gP ,

(2.17) (2.18)

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where CP is the nP × nP diagonal matrix with face areas |fi | on the diagonal and BP = CP 1. The second equation is the local mass balance equation with gP being the mean value of the source term. In the first equation, λP ∈ RnP is the vector of Lagrange multipliers representing mean pressure values on the faces of P. These equations are closed by setting essential boundary conditions (for λP ) and imposing the continuity of the pressure and velocity unknowns (λP and uP ) on the mesh faces. 2.3.1. The elemental mass matrix The main difference between the MFD and MFE methods is in the construction of the local mass matrices MP , which approximates the volume integrals: Z uTP MP vP ≈ K−1 u · v dV, uP = ΠP (u), vP = ΠP (v). P

Once the local matrices are computed, they are assembled into a global matrix. In the MFE method, the local matrix MP is calculated using the explicit interpolation operator IP , similar to what is done in Section 2.2.1: Z uTP MP vP ≈ K−1 IP (uP ) · IP (vP ) dV. (2.19) P

Extension of the MFE method to polygonal and polyhedral meshes requires a non-trivial construction of the interpolation operator. Attempts in this direction are based on a subtriangulation (tetrahedralization in 3D) of the element and the elimination of the internal degrees of freedom through static condensation75, 123, 124 . In the MFD method, the mass matrix is built as a constrained solution of a set of algebraic equations. This construction is similar to that of the stiffness matrix in Section 2.2.1 and is based on an exactness property also called the consistency condition, which we state as follows: for any constant vector-valued field q and any vector-valued field vh ∈ SP it holds that: Z T (ΠP (vh )) MP ΠP (q) = K−1 (2.20) P q · vh dV. P

The vector-matrix-vector product in the left-hand side of (2.20) gives the exact value of the volume integral in the right-hand side of (2.20). This property ensures that the mimetic method is firstorder accurate for the velocity field33 . Of coarse, this can be done only if we imposed some weak restrictions on the space SP . In particular, the normal components of vh ∈ SP are constant on faces f of P and div vh is constant on P. Under these restrictions, by applying the Gauss-Green formula and noting that there exist a linear polynomial ζ(x) such that ∇ζ = K−1 P q and ζ(xP ) = 0, we obtain: Z Z Z Z K−1 q · v dV = ∇ζ · v dV = − ζ div v dV + ζ nP · vh dS h h h P P

P

P

Z = − div vh

ζ dV + P

X f∈∂P

∂P

Z nP · vh

ζ dS = f

X

vf (nf · nP,f ) |f| ζ(xf ).

f∈∂P

Note that this formula also holds for the low-order Raviart-Thomas MFE method on simplicial meshes, since this space satisfies the restrictions imposed on SP . Nonetheless, in the MFE method

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this relation provides a unique scheme. In the MFD method the consistency condition (2.20) leads to a family of schemes. Let us explore this issue in detail. Due to the linearity of the integration, the result has the form of a linear functional of ΠP (vh ). This means that there exists a vector rP (q) ∈ RnP that depends only on the geometry of element P, the constant vector q (we stress this fact explicitly), and the tensor KP such that Z T K−1 ∀vh ∈ SP . P q · vh dV = (ΠP (vh )) rP (q) P

Inserting this relation in (2.20) and observing that ΠP (vh ) is an arbitrary vector in RnP , we obtain: ∀q ∈ (P0 (P))d .

MP ΠP (q) = rP (q)

(2.21)

Due to the linearity of the matrix-vector operations, this gives only d linearly independent conditions; thus, the system is underdetermined when nP > d. To derive a more convenient representation of the matrix equations (2.21), we introduce two rectangular nP × d matrices NP and RP defined as follows. Let qi = ∇xi for i = 1, . . . , d. Then: NP = ΠP (q1 ), . . . , ΠP (qd ) and RP = rP (q1 ), . . . , rP (qd ) . The equations (2.21) are reduced to a single matrix equation: M P N P = RP .

(2.22)

(0) MP

Let denote a particular solution to (2.22). Then, the family of mimetic mass matrices is given by the parametric relation (0)

(1)

MP = MP + MP (α1 , . . . , αk ),

(2.23) (1) MP

which is expressed through the k parameters α1 , . . . , αk . Here, matrix ensures the stability of the MFD method. The total number of parameters is k = (nP − d)(nP − d + 1)/2 and grows quadratically with nP . For example, k = 1 for a triangle and a tetrahedron, k = 3 for a quadrilateral and k = 6 for a hexahedron. 2.3.2. Explicit construction of the mimetic mass matrix The matrix equation (2.22) can be solved using the standards tools of the linear algebra. It is easy to verify that the particular solution is given by (0)

MP = RP (RTP NP )−1 RTP . RTP NP

(2.24)

We will see below that is a nonsingular matrix. This matrix has rank d. The columns of matrix NP are the projections of the vector basis q1 = (1, 0)T , q2 = (0, 1)T in two-dimensions and q1 = (1, 0, 0)T , q2 = (0, 1, 0)T , q3 = (0, 0, 1)T in three dimensions. The corresponding columns of matrix RP are obtained from (2.21). For example, using a local enumeration of faces of a polygon P, e.g., fi for i = 1, . . . , nP , we have the explicit formulas:     nTf1 |f1 |(xf1 − xP )T  nT   |f2 |(xf2 − xP )T   f2    K and RP =  NP =  (2.25)  . . ..  .    . .   nTfn |fnP |(xfnP − xP )T P

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Similar formulas hold in three dimensions. Note that the consistency condition holds true when vh is a constant vector function. Using this fact, a straighforward derivation give the following result36 . Lemma 2.2. Let matrices RP and NP be defined as above. Then, RTP NP = |P|KP . From this lemma, it is readily deduced that NTP RP is a d × d-sized SPD matrix, and, hence, a (1) non-singular matrix. Since the mass matrix MP must be SPD, a proper stability matrix MP has to be constructed to obtain a convergent mimetic scheme. This is done in Section 2.4. 2.3.3. Convergence of the mimetic method ˜ = {vf }f∈Ωh and p ˜ = {pP }P∈Ωh be the vectors collecting all the degrees of freedom of velocity Let v and pressure, respectively. Let Πf and Π be the corresponding global projection operators. We define the following mesh-dependent norms: X X |||˜ v|||2 = vPT MP vP and |||˜ p|||2 = |P| p2P P∈Ωh

P∈Ωh

We have the following convergence result proven in Ref. [33]. Theorem 2.3. Let {Ωh }h be a sequence of regular mesh partitions of Ω, conveniently labeled by the mesh size parameter h for h → 0. Futhermore, let p ∈ H 2 (Ω) and u ∈ H(div, Ω) be the solution ˜ and u ˜ be the solutions of the of (2.13)-(2.14) with the Dirichlet boundary conditions. Finally, let p mimetic scheme. Then, there exists a constant C independent of h such that: |||˜ p − Π(p)||| + |||˜ u − Πf (u)||| ≤ Ch||p||H 2 (Ω) . Under more stringent hypothesis on the shape of Ω and on the regularity of the loading term g, a superconvergence of the pressure variable can be proven33, 89 . (1)

2.4. General form of the stability matrix MP

(1)

The general form of the nP × nP -sized stability matrix MP is given by (1)

MP = DP UP DTP .

(2.26)

The nP × m matrix DP is such that NTP DP = 0 and the columns of NP and DP span RnP , i.e., span{NP , DP } = RnP . Thus, m = nP − d − 1 for the nodal mimetic discretization and m = nP − d for the mixed mimetic discretization. Matrix UP is an (nP − m) × (nP − m)-sized symmetric and positive definite matrix of parameters. In practice, an effective choice for matrix UP that leads to a one-parameter family of schemes is given by UP = α I, where α > 0 and is properly scaled, see Lemma 2.3. Alternatively, we can set (1) UP = α (DTP DP )−1 , where α > 0, which gives MP = α DP (DTP DP )−1 DTP . Now, we observe that DP (DTP DP )−1 DTP + NP (NTP NP )−1 NTP = I,

(2.27)

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M-adaptation

13

since span{NP , DP } = RnP by the construction and the columns of DP are orthogonal to the columns of NP . Thus, we can reformulate this single parameter family as follows (1) MP = α I − NP (NTP NP )−1 NTP . (2.28) Note that the elemental mimetic mass matrix MP is SPD for any choice of the positive parameter α36 . Similarly, the elemental mimetic stiffness matrix is semi-positive definite and has a single kernel vector 1 for any choice of α > 0. However, convergent schemes are obtained only when α is properly (0) scaled. A recommended choice is α = trace(MP )/nP . (0)

Lemma 2.3 (Stability condition). Let α = trace(MP )/nP and Assumption 2.1 hold. Then, (0) (1) the mimetic mass matrix MP = MP + MP is spectrally equivalent to the scalar matrix |P| I, i.e. ∗ there exists two constants σ∗ and σ , which are independent of P and of the mesh Ωh , such that σ∗ |P| uTP uP ≤ uTP MP uP ≤ σ ∗ |P| uTP uP

∀uP .

(2.29)

3. Maximum principles for nodal MFD method 3.1. Theoretical background Let us recall the fundamental results from the theory of elliptic PDEs concerning the strong and the weak 53 maximum principles.

53, 102

Theorem 3.1 (Strong maximum principle). Let us assume that function u(x) satisfies − div(K∇u) ≤ 0

in

Ω.

Furthermore, let K be a strongly elliptic tensor in Ω and Ω be regular enough. If u attains a nonnegative maximum umax at an interior point of Ω, then: u = umax

in Ω.

Theorem 3.2 (Weak maximum principle). Let us assume that function u(x) satisfies − div(K∇u) ≤ 0

in

Ω.

Furthermore, let K be a strongly elliptic tensor in Ω and Ω be regular enough. Then, max u(x) ≤ max 0, max u(x) . x∈∂Ω

x∈Ω

From the weak maximum principle it is immediate to derive a monotonicity property for the Dirichlet boundary value problem. In the case of mixed boundary conditions, the monotonicity property formulated as follows 70 . Corollary 3.1 (Monotonicity property). Let us assume that function u(x) satisfies − div(K∇u) ≥ 0

in

Ω,

n · K∇u ≥ 0

on

ΓN ,

u≥0

on

ΓD = ∂Ω\ΓN .

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4

θ bj T

d1 T4 3 T 3 d2 T2

dj Tj 1

vj

2

b j associated Fig. 2. The left picture shows the diagonal vector dj in a convex quadrilateral and the triangles Tj and T with the vertex vj . The right picture shows the diagonal vectors d1 and d2 and the triangels T2 , T3 , and T4 in a concave quadrilateral. The triangle T1 (not shown) is defined by vertices 1, 2, and 4. Note that the oriented area of T3 is negative. In both pictures, the subscript j runs from 1 to 4 counter clock-wise.

Furthermore, let K be a strongly elliptic tensor in Ω and Ω be regular enough. Then, u≥0

in

Ω.

By reversing the inequalities of the above statements, we obtain the strong and weak mimimum principles. Hereafter, the maximum principle referes to both the maximum and minimum principles. 3.2. The nodal mimetic discretization on quadrilateral cells The widely used approach to achieve the DMP in a numerical scheme is to show that the matrix of the resulting linear system is an M-matrix. The argument presented in Refs. [81, 82] states: If the matrix of system (2.4) is an M-matrix with strong diagonal dominance in at least one row and weak diagonal dominance in the other rows, then the numerical solution satisfies the DMP. It is easy to show that the local sufficient conditions yielding this property imply that all the elemental stiffness matrices MP are M-matrices. These local conditions are not always necessary but they are much easier to control in practice. The goal of the m-adaptation is to enforce these local conditions via a proper selection of the parameters. Following Ref. [96], for a quadrilateral cell P, we define the following geometric objects (see Fig. 2 for detail): four oriented diagonal vectors dj , j = 1, . . . , 4, four related triangles Tj , and their b j , i.e. T b j = P\Tj . Due to this definition, d3 = −d1 and d4 = −d2 , so that we four complements T b 1 = T3 , T b 2 = T4 , T b 3 = T1 , and can express all results in terms of d1 and d2 only. Furthermore, T b b T4 = T2 . The oriented areas of the subcells Tj and Tj are denoted by the same symbols Tj and b j , respectively. Thus, T b1 + T b3 = T b2 + T b4. |P| = T b i is always negative when P is non-convex. Note that one and only one of the four oriented areas T

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M-adaptation

15

3.2.1. M -matrix conditions e P be the rotated permeability tensor KP : Let K e P = RT KP R K

with R =

01 −1 0

.

(3.1)

As it was mentioned above the total number of parameters is k = (nP − d − 1)(nP − d − 2)/2, so in the case of quadrilateral we have a one parameter family. The following technical result is proved in Ref. [96]. Lemma 3.1. The ij-entry in the stiffness matrix MP has the following representation: MP

(0)

ij

(1)

= (MP )ij + (MP )ij =

b b 1 e P dj + (−1)i+j Ti Tj α1 , di · K 4|P| |P|2

(3.2)

where α1 is a non-negative parameter. Under a few assumptions, which depend only on the shape of P and the permeability tensor KP , there exists a range of values for the parameter α1 such that MP is an M-matrix. Due to the (0) geometric relations, the consistency term MP of MP takes a very peculiar block-structured form: ! e P d1 d1 · K e P d2 1 S −S d1 · K (0) where S = MP = e P d2 d2 · K e P d2 . 4|P| −S S d1 · K Theorem 3.3. (i) Let P be a convex quadrilateral cell. Then, matrix MP is an M-matrix if the parameter α1 satisfies the two inequalities ( ) ) ( e P d2 e P d1 d2 · K e P d2 e P d2 d1 · K d1 · K 4α1 −d1 · K ≤ min , . (3.3) max , ≤ b1T b1T b4, T b2T b2, T b3T b1T b2T b 3 ) min(T b4) b3 b4 |P| min(T T T A sufficient condition ensuring that the set of possible values of α1 satiosfying (3.3) is non-empty is given by: n o e P d2 ≤ 4 min d1 · K e P d1 , d2 · K e P d2 . d1 · K (3.4) (ii) Let P be a non-convex quadrilateral cell. Then, there does not exist any parameter α1 for which MP is an M-matrix. Proof. Formula (3.3) is a sufficient condition for both convex and non-convex quadrilaterals that follows from the representation (3.2). The second statement is trivial. If P is a non-convex b1T b 3 < 0 or T b2T b 4 < 0. In such a case, the left-most term in inequality (3.3) is cell, then either T non-negative while its right-most term is strictly negative, since otherwise P would be degenerate. Thus, there exist no α1 > 0 satisfying both inequalities. 2 Following Ref. [96], we illustrate this theorem by presenting admissible quadrilateral cells for which the mimetic family contains an M-matrix. Let KP be the identity matrix. As shown in Figure 3, we fix three vertices A, B and C and vary the fourth vertex D. If we choose this vertex inside the grey region, we obtain the quadrilater cell ABCD for which the set of values for α1

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C

1

B

θ 0.5

A

D

0

!0.5

!1

!1.5

!2

D0 !2

!1.5

!1

!0.5

0

0.5

1

1.5

Fig. 3. A sketch of the monotonicity region (in grey) for KP = I and three three fixed vertices A, B, C.

1

1

0.5

0.5

0

0

!0.5

!0.5

!1

!1

!1.5

!1.5

!2

!2

!1.5

!1

!0.5

0

0.5

!2 1

1

!2

!1.5

!1

!0.5

0

0.5

1

1.5 3

0.5 1

2

0.5

1

0

0

!0.5

!0.5 0

!1 !1

!1

!2

!1.5

!1.5

!2

!1

0

1

2

3

4

!2

!2

!1.5

!1

!0.5

0

0.5

1

1.5

2

!2

!2

!1.5

!1

!0.5

0

0.5

1

1.5

\ which take values 147◦ , 119◦ , and Fig. 4. Monotonicity regions (in grey) for KP = I and different angles θ = ABC 98◦ in the top row, and 20◦ , 47◦ and 90◦ ,in the bottom row.

. specified by inequalities (3.3) is non-empty. If we choose the fourth vertex outside the grey region, e.g., point D0 on the picture, we obtain the quadrilater cell ABCD0 for which inequalities (3.3) cannot be satisfied. We will refer to the gray region as the “monotonicity region”. \ and the value of The shape of the monotonicity region strongly depends on the angle θ = ABC the diffusion tensor KP , as shown by Figures 4-5. It is worth noting that in the presented sequence, where θ varies from an obtuse angle to an acute angle, an infinite part of the ray originating at point B, orthogonal to segment AC, and lying outside the triangle ABC is always inside the monotonicity region. In fact, for points D on this ray, the left-most term in inequality (3.3) is always zero while the right-most term is always strictly positive; hence, the set of values of α1 satisfing (3.3) is never empty. For very acute angles θ, the monotonicity region tends to an infinite strip whose base is the segment AC.

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M-adaptation 1.5

1.5

1

1

1

0.5

0.5

0.5

1.5

0

0

0

−0.5

−0.5

−0.5

−1

−1

−1

−1.5

−1.5

−1.5

−2

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2

1.5

1.5

1

1

1

0.5

0.5

0.5

0

0

0

−0.5

−0.5

−1

−1

−1

−1.5

−1.5

−1.5

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

1.5

−0.5

−2

17

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2

\ as in Figure 4. The full Fig. 5. Monotonicity regions (in grey) for a full tensor KP and the same angles θ = ABC tensor rotates the monotonicity region. Note the existence of a limiting angle beyond which the monotonicity region is empty.

The monotonicity regions of Figure 4 are computed by taking KP = I, while those of Figure-5 corresponds to a full diffusion tensor. By comparing Figures 4 and 5 we note that the effect of the diffusion tensor is to rotate the monotonicity region. 4. Maximum principles for mixed MFD method Efficient solvers for the mixed mimetic discretization use a hybridization technique that introduces two fluxes on each interior mesh face and adds trivial equations expressing their continuity. Since this technique uses only M−1 P , we have to parametrize this matrix. In the mimetic framework, we can directly build a matrix WP such that NP = WP RP .

(4.1)

(compare equation (4.1) with equation (2.8)). In view of the developments of Section 2.4, a general formula for WP is given by WP = NP (NTP RP )−1 NTP + DP UP DTP ,

(4.2)

where DP is a maximum rank nP × (nP − d)-sized matrix such that DTP RP = 0, and UP = UP (α1 , . . . , αk ) is an (nP −d)×(nP −d)-sized matrix of parameters. With a small abuse of notation, we use the same symbols DP and UP as in Section 2.4. Matrix WP is the inverse of matrix MP in the following sense. Once a particular matrix MP has been selected, there exists a choice of parameters αi that determines a matrix WP such that WP = M−1 P . For this reason, we have the following results proved in Ref. [36]. Lemma 4.1. (i) Let matrix WP be given by formula (4.2). If UP is an SPD matrix, then WP is also an SPD matrix.

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(ii) Let Assumption 2.1 hold, UP be spectrally equivalent to the scalar matrix |P|−1 I and the columns of DP have unit Euclidean norms. Then, matrix WP is spectrally equivalent to the scalar matrix |P|−1 I, i.e. there exists two constants σ∗ and σ ∗ , which are independent of P and of the mesh Ωh , such that σ∗ T σ∗ T uP uP ≤ uTP WP uP ≤ u uP |P| |P| P

∀uP .

(4.3)

Remark 4.1. Let us rewrite formula (2.17) as uP = WP CP (pP 1 − λP ) (note that BP = CP 1). This relation can be characterized as a multipoint flux approximation formula. Let the boundary of P be formed by nP faces fi , i = 1, . . . , nP , with measures |fi |. Then, a more detailed flux formula is     uf1 |f1 |(λf1 − pP )  uf2   |f2 |(λf2 − pP )      (4.4)  ..  = −WP  . ..  .    . ufnP |fnP |(λfnP − pP ) Each individual flux is a linear combination of the finite differences of the pressure unknowns. 4.1. Mathematical requirements for the DMP Let us discuss a set of sufficient conditions for the matrix WP that lead to a mimetic scheme with the DMP. We assume that the pressure g D (x) (the Dirichlet boundary condition) is set on a part of the domain boundary ΓD and the flux g N (x) (the Neumann boundary condition) is set on the ¯ D . We recall the two assumptions proposed in Ref. [81]. remaining boundary ΓN = ∂Ω\Γ P Assumption 4.1. Let WP = (wij )ni,j=1 . We assume that

(A1) matrix WP satisfies the geometric constraint: X wii |fi | + wij |fj | ≥ 0

∀i,

j6=i

and the inequality is strict for at least one matrix row; (A2) matrix WP is a Z-matrix, i.e., wij ≤ 0 for i 6= j. From Assumption 4.1 it immediately follows that matrix WP is a Stiltjes matrix and so is also an M-matrix 20 . This fact makes it possible to prove the following theorems. Theorem 4.2 (Discrete maximum principle). Let (pP )P∈Ωh and (λf )f∈Ωh be the solutions of the hybrid mimetic method under Assumption 4.1. Furthermore, let g(x) be a nonnegative source function in Ω and g D (x) and g N (x) be nonnegative functions on the Dirichlet, ΓD , and the Neumann, ΓN , boundaries, respectively. Then pP ≥ 0 and λf ≥ 0 for any P and any f. Theorem 4.3 (Discrete minimum principle). Let (pP )P∈Ωh and (λf )f∈Ωh be the solutions of the hybrid mimetic method under Assumption 4.1. Furthermore, let g(x) be a nonpositive source function in Ω and g D (x) and g N (x) be nonpositive functions on the Dirichlet, ΓD , and the Neumann, ΓN , boundaries, respectively. Then, pP ≤ 0 and λf ≤ 0 for any P and any f.

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M-adaptation

19

Using both Theorems 4.2 and 4.3, we obtain a discrete version of the weak Hopf’s Lemma (e.g., Theorem 3.2). Theorem 4.4. Let (pP )P∈Ωh and (λf )f∈Ωh be the solutions of the hybrid mimetic method under Assumption 4.1. Furthermore, let g(x) = 0 and ΓN = ∅. Then, the values of pP and λf are bounded by the maximum and minimum mean values of g D (x) on the boundary mesh faces. 4.2. Analysis of particular cases Assumption 4.1 combined with the requirement of the positive definiteness of matrix UP (α1 , . . . , αk ) lead to a set of inequalities for parameters αi . We present the cases where these inequalities can be analyzed theoretically and analytic solutions can be found. In general, they can be solved numerically. 4.2.1. Triangular and tetrahedral meshes For a simplex P we have nP = d + 1; hence, UP is a 1 × 1 matrix, UP = α1 . The interesting property of a simplex is that the Lagrange multipliers λf do not depend on the parameter α1 . However, solutions pP do depend on α1 . Formula (4.2) gives81 : wij =

1 T 1 n K nfj + α1 . |P| fi |fi | |fj |

Assumption (A1) is reduced to α1 > 0. Since α1 is arbitrary, the analysis of the limiting case α1 → 0 allows us to extend the monotonicity result to the right angles in the case of an isotropic tensor KP and to condition nTfi K nfj ≤ 0 in the case of a full tensor. Assumption (A2) is satisfied if nTfi K nfj < 0 and this condition is consistent with the similar well-known results for the RaviartThomas MFE method. In fact, the family of mimetic scheme on simplicial meshes contains this MFE method41 . In conclusion, on simplicial meshes, our analysis recovers the well established angle condition. In particular, when KP is an isotropic tensor, the M-matrix requirement is that the angles between faces (dihedral angles in three dimensions) are less than 90◦ . 4.2.2. Non-simplicial meshes In contrast to simplicial meshes, meshes of general polygonal and polyhedral cells have more freedom for constructing a discretization scheme with the DMP. In formula (4.2), the matrices RP and NP have explicit representation while the matrix DP is defined implicitly by RTP DP = 0 and is not unique. The lack of an explicit expression for the entries of DP in the case of general cells make the analysis of assumptions (A1) and (A2) rather involved. Therefore, we start with the simplest polygonal and polyhedral cells and then we extend our analysis to more general cells via an affine transformation. Square and cubic cells. The geometry of a unit square cell P and the special ordering of its edges shown in Fig. 6 result in a simple structure of matrices RP and NP which in turn leads to an

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V. GYRYA, K. LIPNIKOV, G. MANZINI and D. SVYATSKIY 3

2

2 1

1

3

4

5

z y

y

x

x

4

6

Fig. 6. The unit square cell (left) and the unit cubic cell (right) with the prescribed edge/face order.

explicit formula for matrix DP :     1 0 1 0  0 1 1 1 0 1 I I 1010 T     = RP =  , NP =  KP = KP , DP = = (I I) , −1 0  −I 0101 2 −1 0  2 −I 0 −1 0 −1 where I is the 2 × 2 identity matrix. Inserting these matrices into formula (4.2), we can write the matrix WP in the 2 × 2 block form: ! KP + UP − KP + UP WP = . (4.5) −KP + UP KP + UP With this structure, the analysis of assumptions (A1) and (A2) becomes simple. Since the length of all edges of P is equal to one, the first assumption is equivalent to: UP 1 > 0, which implies that UP has diagonal dominance in rows. The analysis of individual entries in formula (4.5) implies that the second assumption is satisfied if ! K11 −|K12 P P | , UP ≤ 22 −|K12 P | KP where Kij P are the entries of tensor KP . One of the possible choices for UP which apparently minimizes the sparsity pattern of the resulting discrete system is ! K11 −|K12 P P | UP = . (4.6) 22 −|K12 P | KP Using similar arguments, we can derive the sufficient conditions for a cubic cell, see Ref. [81]. The matrix of parameters for a cubic cell should satisfy the following two inequalities:   13 K11 −|K12 P P | −|KP |   22 UP 1 > 0 and UP ≤  −|K12 (4.7) −|K23 P | KP P |. 23 33 −|K13 P | −|KP | KP

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M-adaptation

21

5

y 1 2

3 1 1

3

2

1 0

1

1

x

2

4 Fig. 7. An orthogonal triangular prism (left) with the right triangle as the base (right). The order of faces is described by circled numbers.

Note that unlike the two-dimensional case, it is not always possible to satisfy these conditions. In fact, it is not difficult to find a full positive definite diffusion tensor KP such that the matrix in the right hand side of inequality is indefinite. Orthogonal prism cell . Another cell type widely used in engineering applications is a triangular prism. As the basic cell we consider the orthogonal prism P shown in Fig. 7. The base of this prism is the right triangle with area 0.5 and its height is 2, so that |P| = 1. The prescribed faces order is also shown in Fig. 7. In addition, we assume that tensor KP satisfies the following inequalities: nTfi KP nfj < 0

i 6= j

and

i, j = 1 . . . 3.

(4.8)

This means that the bottom and top triangular faces satisfy the monotonicity conditions described in Section 4.2.1. These restrictions are inspired by porous media applications where the diffusion tensor is often aligned with geological layers and isotropic in the horizontal directions. For this cell, the explicit formulas for matrices RP and NP are   2   −1 0 0 −3 0 0    0 −1 0   0 − 23 0     2 2  √ √  2 2    RP =  3 3 0  , NP =  2 2 0  KP .    1 0 0 −   0 0 −1  2 0 0 12 0 0 1 The simplest choice of matrix DP such that RTP DP = 0 is as follows: ! 1 1 1 0 0 T DP = 0 0 0 1 1 Recall that nP = 5 and d = 3; hence, UP is a 2 × 2 matrix. We parametrize this matrix as follows: ! α1 α2 UP = with α1 > 0 and α1 α3 − α22 > 0. α2 α3

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V. GYRYA, K. LIPNIKOV, G. MANZINI and D. SVYATSKIY (0)

(1)

According to formula (4.2), matrix WP is the sum of two terms, i.e., WP = WP + WP (α1 , α2 , α3 ). Using the above formulas, we obtain √   2 12 11 12 K11 K − K13 −K13 P P P P 2 (KP + KP ) √   12 22   K22 − 22 (K12 K23 −K23 P P P P + KP )   √ KP √ √ √  (0) 2 2 2 1 12 12 22 12 11 22 13 23 13 23  WP =  − 22 (K11 + K ) − (K + K ) K + (K + K ) (K + K ) (K + K ) − P P P P P P P P P P P  2 2 2   √2   2 13 23 13 23 33 33 (K + K ) K K − K −K   P P P P P P √2 2 13 23 33 33 23 (K + K ) −K K −K13 −K P P P P P P 2 and 

α1 α1 α1 α2 α2



   α1 α1 α1 α2 α2    (1)  WP (α1 , α2 , α3 ) =   α1 α1 α1 α2 α2  .    α2 α2 α2 α3 α3  α2 α2 α2 α3 α3 Now, the analysis of entries of WP as required by Assumption 4.1 leads to the following sufficient conditions on the parameters α1 , α2 and α3 :  23 α2 ≤ min −|K13  P |, −|KP | ,   √ −2(2 + 2)α2 ≤ α3 ≤ K33 (4.9) P , n o    − 1√ α ≤ α ≤ min √2 |K12 + K11 |, √2 |K12 + K22 | . 2(2+ 2)

2

1

2

P

P

2

P

P

This system of inequalities has a solution only if tensor KP satisfies some additional conditions:  √ 23 33  + 2) max |K13  2(2√ P |, |KP | ≤ KP , 12 (4.10) 2 23 11 12 22  √ max(|K13  P |, |KP |) ≤ min |KP + KP |, |KP + KP | . 2(2 + 2) If tensor KP is diagonal, than the above conditions are always satisfied and UP can be defined as follows: ! √ 2 22 min{K11 0 P , KP } 2 UP = . 0 K33 P 4.2.3. Affine transformation Let us analyze how the derived sufficient conditions are changing in the case of an affine transforˆ of a new mation given by matrix T . This matrix transforms a point x of a cell P into the point x b cell P: ˆ = T x. x

(4.11)

The affine transformation preserves the barycenters of any weighted collections of points. It means that the barycenter xP and the barycenters of faces xf of P are moved to the corresponding

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M-adaptation

23

b Lengths, areas and volumes are not preserved under the affine transformation. barycenters of cell P. To make our derivations easy to understand, we introduce diagonal matrices FP and FPb with face b respectively, on the diagonals. Now, we can areas (edge lengths in two dimensions) of P and P, express matrix RPb in terms of RP as follows:     (xbf1 − xPb )T (xf1 − xP )T  (x − x )T   (x − x )T  b  T  f2  bf2  P P  T = Fb F−1 RP T T = Λ−1 RP T T ,    (4.12) RPb = FPb  .. P P   = FPb  ..     . . (xbfn − xPb )T (xfnP − xP )T P

FP F−1 b . P

where Λ = It is easy to check that vector T −T nfi is orthogonal to a new face if vector nfi is orthogonal to an old one. The norm of this vector is not equal to one, so it has to be scaled and the relationship between the two normal vectors is defined as follows: 1 nbfi = T −T nfi . (4.13) kT −T nfi k Now we have to prove the following Lemma. Lemma 4.2. Let the affine transformation be given by matrix T . Then, |bfi | = | det T | kT −T nfi k. |fi | Proof. We show this result in three dimensions. In two dimensions, the proof is similar. Let us consider a face fi with its unit normal vector nfi . We define an infinitesimal volume dv as the orthogonal prism with the base ds and the unit height, so that |dv| = |ds|. The affine transformation c with the base ds. b The defined by matrix T transforms this volume to an infinitesimal volume dv b new volume dv is in general a tilted prism. Its volume is calculated as follows: c = (nb · T nf )|ds| b = |dv| i fi

1 kT −T nfi k

b = (T −T nfi · T nfi )|ds|

b |ds| kT −T nfi k

.

c = | det T ||dv|. Hence, It is known that the affine transformation scales volume as | det T |, i.e. |dv| b |ds| = | det T | kT −T nfi k |ds| and the assertion of the lemma follows immediately. Using formula (4.13) and definition of matrices NP and NPb , we obtain −1 NPb = | det T | Λ NP K−1 KP . P T

The next step is to derive a matrix DPb , such that RTPb DPb = 0. Let us suppose that matrix DP , such that RTP DP = 0, is given. Straightforward derivations result in: DPb = Λ DP .

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Fig. 8. Examples of cells which can be obtained by affine transformation of basic cell types.

Using the above derivations, we can write down a new formula for matrix WPb : WPb =

1 1 NPb K−1 NTPb + DPb UPb DTPb = Λ(NP K−1 )KT (K−1 NTP )Λ + ΛDP UPb DTP Λ P b |P| |P|

= ΛWP (KT )Λ,

(4.14) (4.15)

where KT = | det T |T −1 KP T −T . Let WP (KT ) denote the inverse of the mass matrix for cell P and the modified diffusion tensor KT . b which can be obtained from a basic cell type discussed in Let us consider the family of cells P the previous subsection (square, cube or orthogonal prism) by affine transformations (4.11). In the e In the case of the unit cube, case of the unit square P, the family contains the parallelograms P. it contains oblique parallelepipeds, see Fig. 8. Now, we derive sufficient conditions which guarantee that assumptions (A1) and (A2) hold for b Assumption (A1) can be written in the following form: cell P. WPb FPb 1 ≥ 0. Let multiply both sides of this inequality from the left by matrix Λ−1 . This does not affect the solution, since each row is scaled by a positive number. Using formula (4.14), we obtain: Λ−1 WPb FPb 1 = WP (KT )FP 1 ≥ 0, which is equivalent to the assumption (A1) for the basic cell with diffusion tensor KT . Thus, for both parallelograms and oblique parallelepipeds the sufficient condition is the same as for the corresponding basic cell types, namely: UPb 1 > 0. Assumption (A2) is trivial. Indeed, it is easy to show that WPb is a Z-matrix if and only if matrix Λ−1 WPb Λ−1 = WP (KT ) is a Z-matrix. It means that we can utilize the sufficient conditions derived for the basic cell types taking into an account that they have be to applied with the modified diffusion tensor KT . For example, for a parallelogram we have: ! K11 −|K12 1 T T | , UPb ≤ b −|K12 | K22 |P| T

T

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and for an oblique parallelepiped:  UPb ≤

K11 T

13 −|K12 T | −|KT |



 1   −|K12 | K22 −|K23 |  . T T T   b |P| 23 33 −|K13 T | −|KT | KT

where Kij T are the entries of tensor KT . We recall that it is not always possible to find a positive definite matrix UPb which satisfies both conditions. 5. Acoustic wave equations Consider a problem of efficient and accurate numerical solution of an acoustic wave equation in two dimensions59, 60 and three dimensions61 utt = c2 4u,

(5.1)

where u is a scalar displacement and c is the wave speed. For simplicity, we consider periodic boundary conditions. To discretize this equation, we consider a space Vh of nodal-based mesh functions. Let uk ∈ Vh be the discrete vector of degrees of freedom at time tk = k 4t. Then, the mimetic discretization of (5.1) has the form M

uk+1 − 2uk + uk−1 = Auk , 4t2

(5.2)

where the global mass matrix M and the global stiffness matrix A are derived from the elemental matrices: X X M= NP MP NPT and A= NP AP NPT . P∈Ωh

P∈Ωh

The construction of the stiffness matrix AP has been described in Section 2. The construction of the mass matrix MP will be described later. One of the essential steps to achieve efficiency of the numerical scheme is to simplify the solution of a linear system performed at every time step. In the finite element community one technique that does this is called mass lumping, where the off-diagonal entries of the mass matrix M are moved to the diagonal to obtain a diagonal matrix D. Another approach 57, 129 is to approximate the inverse of the mass matrix M−1 ≈ W := D−1 M D−1 . In addition to simple inversion of the mass matrix, this approach is preferred for the m-adaptation, as it leaves more freedom. Both, the mass and stiffness matrices are described by a set of parameters. The final scheme has the form uk+1 − 2uk + uk−1 = W Auk . 4t2

(5.3)

Just like the local stiffness matrix AP , the local matrix mass matrix MP is derived based on a consistency condition. For the acoustic equations, the consistency requirement is weaker 47 : For

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any constant function v 0 the vector-matrix-vector product gives the exact value of the following integral: Z T 0 u v 0 dV ∀u ∈ SP , (5.4) ΠP (u) MP ΠP (v ) = P

where SP is the space of continuous functions that are integrated exactly by any quadrature rule with the quadrature point at vertices of cell P. Taking the simplest quadrature rule with equal weights, we obtain the following algebraic form of the consistency condition: |P| 1, nP where nP is the number of vertices in P. It is easy to check that the general solution of this problem is given by MP 1 =

(0)

(1)

MP = MP + MP (α1 , . . . , αk ),

(0)

MP =

|P| 1 1T , n2P

(1)

MP

1 = 0,

(1)

where k = nP (nP − 1)/2. For a quadrilateral cell, matrix MP is described by six parameters. For a cuboid cell, the number of parameters grows to 28. For each selection of parameters that satisfy a stability condition, we obtain a scheme that is second-order accurate in space and time. Next we present the optimization of the parameters in the mass and the stiffness matrix so as to minimize the numerical dispersion and the numerical anisotropy of the scheme. The optimization criteria can be formulated in a number of ways. In the subsequent sections, we present two approaches: (i) based on the conventional von Neumann-type analysis 59, 60 and (ii) based on the cancelation of error terms coming from the spatial and the temporal discretizations. 5.1. M-adaptation based on von Neumann-type analysis Let us consider a plain exponential wave u(x, t) = exp {κ · x − ch κt} ,

κ = |κ|,

(5.5)

such that vector Π(u) is the solution of (5.3). For the numerical schemes, typically, the numerical wave speed ch 6= c, but ch = ch (κh) asymptotically approaches c for finely resolved waves, kh → 0. By substituting the wave form (5.5) into (5.3) one obtains a dispersion relation (dependence of the numerical wave speed ch on the discretization parameters and the wave number k). In [60] it was shown that this dependence can be drastically simplified for a rectangular mesh. Let P be a rectangle with sides 4x and 4y and the parameters be the same in each rectangle of the mesh. Then, the dispersion relations can be written in the following compact form: 2(1 − cos(ch 4t)) = c2 4t2 (v∗ WP v)(v∗ AP v),

(5.6)

−1 where WP = D−1 P MP DP and v is a vector assembled from values of (5.5) at the four vertices (x1 , . . . , x4 ) of the reference rectangle P (for t = 0),   iκ·x1   e 1  eiκ·x2    eiκ2 4y    x1 = (0, 0) and κ = (κ1 , κ2 ), (5.7) v=  eiκ·x3  = e−iκ1 4x e−iκ2 4y , eiκ·x4 eiκ1 4x

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and v∗ is its complex conjugate transform. The dispersion relation written in the form (5.6), in particular, is convenient for identifying the stability conditions for the scheme, i.e. the numerical velocity ch takes real values for all wave numbers κ. This means, on one hand, that the matrices WP and AP have to be non-negative definite and, on the other hand, the time step has to satisfy the stability condition 2 c4t ≤ kvk−4 kWP k−1 kAP k−1 = 4−2 kWP k−1 kAP k−1 (5.8) h in two dimensions. In three dimensions the stability condition can be written in a similar form.

Fig. 9. The dependence of the numerical wave speed ch on the wave resolution parameter κh for various directions of the wave, κ/κ, relative to the mesh for the Courant number ν = 3/4 for the modified quadrature scheme (left) and the m-adaptation scheme (right). Both schemes have a fourth order numerical dispersion in κh, but the madaptation scheme has sixth order numerical anisotropy (vs forth order). The smaller anisotropy manifests itself through smaller dependence of the dispersion curves on the orientation of the waves relative to the mesh.

The dispersion relation (5.6) can be expanded as follows: ch = c + c1 κh + c2 (κh)2 + . . .

(5.9)

The dependence of the coefficients ci on the parameters can be made explicit by substituting the expansion (5.9) into the dispersion relation (5.6) and expanding both sides in orders of κh 1. Next, we select the parameters so as to cancel the leading terms c1 , c2 , etc., in this expansion (5.9). For a rectangular mesh, it is possible to eliminate the numerical dispersion up to the fourth order59 in κh 1. For a square mesh, additionally, it is possible to eliminate the numerical anisotropy up to the sixth order59 in κh 1, i.e. c1 = · · · = c3 = 0 in (5.9) and c4 , c5 are independent of the direction κ/κ of the wave. This is illustrated in Fig. 9. In all of this cases we are improving accuracy of the numerical velocity for small values of κh, i.e. finely resolved waves. Higher order of accuracy implies accuracy for sufficiently large values of κh. For example, κh = 1 implies that we can use approximately six mesh points per wavelength.

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5.2. M-adaptation based on cancellation of discretization errors For a mesh of cuboids (analogs or rectangles in three dimensions) the approach described above is no longer tractable as the number of parameters of the MFD scheme becomes too large (28 parameters for the mass matrix and ten parameters for the stiffness matrix). A different approach was devised in Ref. [61], where instead of considering an exponential wave (5.5) authors considered plane polynomial waves Pκd (x; t) = (κ · x − ct)d ,

d = 1, . . . , dmax .

(5.10)

The criteria for minimizing the numerical dispersion in this approach are to cancel the temporal and the spatial discretization errors for all waves (5.10) for d = 1, . . . , dmax , i.e. Etime Pκd (x; t) v = Espace Pκd (x; t) v , (5.11) where the errors (that are vectors from Vh ) are defined as Espace [u]

v

= − W A uk

+ ∆u(xv ), v

Etime [u]

v

=

uk+1 − 2ukv + uk−1 ∂2 v v − u(xv ) (5.12) 4t2 ∂t2

and evaluated at every point xv and every time step tk . The degree dmax defines the order of the approximation. Note, that the cancellation of errors conditions (5.11) yields an infinite family of conditions, as they have to hold for any point xv in space and time tk and for any direction κ. Therefore, it is difficult to say if it is even possible to satisfy all of these conditions simultaneously. There are two observations that drastically simplify the analysis of the conditions (5.11). The first observation is that it is sufficient to check the conditions (5.11) at the origin (xv = 0, tk = 0) only. If the conditions at the origin are satisfied, then the conditions at all other points (xv , tk ) are also satisfied. Lemma 5.1. Suppose the conditions (5.11) are satisfied at the origin, then they are satisfied for any point (x0 , t0 ). Proof. Every plane polynomial wave (5.10) can be written in terms of a linear combination of plane polynomial waves of the same degree and lower, written in the coordinates shift by (x0 , t0 ), by using a finite Taylor expansion in (κ · (x − x0 ) − c(t − t0 )): X ˜ Pκd (x; t) = Pκd (x − x0 ) + x0 ; (t − t0 ) + t0 = Cd˜Pκd x − x0 ; t − t0 . ˜ d≤d

Thus, due to the linearity of the error operators Etime and Espace , we have Espace Pκd (x; t) v − Etime Pκd (x; t) v = n h h X i i o ˜ ˜ Cd˜ Espace Pκd x − x0 ; t − t0 − Etime Pκd x − x0 ; t − t0 . ˜ d≤d

v

(5.13)

v

Since the right side of (5.13) is zero (by assumption of the Lemma), then so is the left side. This proves the assertion of the lemma.

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The second observation simplifying the analysis is that the conditions (5.11) for the infinite set of directions κ can be reduced to an equivalent finite set of conditions on the monomials pr in x, y, and z of degrees two and four (evaluated at the origin ˜v due to Lemma 5.1) of the form: Espace [pr ] ˜v = w˜vr , r = 1, . . . , rmax , (5.14) The spatial error w˜vr is zero (trivial) for most of the monomials. It is nontrivial for Espace x4 ˜v = w˜vr = 2c2 4t2 , Espace x2 y 2 ˜v = w˜vr = 23 c2 4t2 ,

(5.15)

and other monomials obtainable from (5.15) by permutations of variables x, y and z.

Fig. 10. The relative error in the numerical wave speed ch as a function of the wave resolution parameter κh for various directions of the wave, κ/κ, relative to the mesh for the Courant number close to ν ≈ 1/2 for the extension of the modified quadrature scheme to cubes (left) and the m-adaptation scheme on cuboids (right).

There are six conditions for six monomials of degree two and 15 conditions for monomial of degree four. The monomial of order three produce trivial conditions due to the symmetry of the mesh. The total number of parameters is 38, which may be sufficient to satisfy all 21 conditions. The appropriate parameters can be found numerically61 . One has to be careful to take into account the stability conditions (5.8). To maximize the stable time step, given by this formula, one needs to minimize the norms of the matrices WP and AP . Let pr = Π(pr ) be the projection of pr on the discrete space. Then, the problem that has to be solved numerically is a constrained minimization problem: n o min kWP k, kAP k subject to W A pr ˜v = wr , r = 1, . . . , 21. (5.16) In Ref. [61] a solution to (5.16) was found for cubic and cuboid meshes. The relative error in the numerical velocity ch is compared in Fig. 10 for the m-adaptation (on a cuboid mesh) and the modified quadrature57 (on a cubic mesh) approaches. The relative error is shown for all directions κ/κ of the wave and various wave resolution parameters κh. The fourth order of the dispersion for both methods can be deduced from the slope of the curves in Fig. 10. One can also observe a smaller numerical anisotropy (dependence of the

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numerical velocity ch on the direction κ/κ of the wave) for the m-adaptation scheme, as seen in the smaller spread of the curves. Note that the modified quadrature approach, which is considered the state-of-the-art in this field, loses accuracy on general cuboid meshes. 6. Multigrid solvers Algebraic multi-grid (AMG) methods are algorithms for accelerating the numerical solution of PDEs using a hierarchy of coarser and finer discretizations in iterative schemes. They are commonly applied for elliptic problems and are among the-state-of-the-art techniques. The main goal of this section is to explore the possibility of further improving the performance of the AMG methods using m-adaptation technique. Since the the two most expensive steps in numerical solution of a PDE are (i) the generation of the mass and the stiffness matrices and (ii) the iterative solution of the algebraic problem, we will be interested in the overall performance improvement. That is, the performance improvements should offset the additional computational time required for the optimization of the matrices. 6.1. M-adaptation for AMGs The theory of AMGs, originally, has been developed for M-matrices and most of the existing open source and commercial solvers show the best performance for such matrices. The family of mimetic schemes contains schemes that lead to an algebraic problems with M-matrices when the mesh and problem coefficients satisfy certain conditions, see Sect. 3 and 4. In the case of a general polygonal/polyhedral mesh, it is not known if the mimetic scheme contains an M-matrix and the search for such a matrix can be computational expensive. Therefore, other optimization strategies are needed. AMG methods show often poor convergence properties for matrices with large positive off-diagonal elements. An inexpensive indirect way to avoid such matrices is to increase the diagonal dominance of the mass matrix WP derived in Sec. 4. Let the mass matrix WP = {wij }ni,j=1 be parameterized as in (4.2): (0)

(2)

WP = WP + WP (α1 , . . . , αk ),

where

(2)

WP = DP UP DTP

and UP = UP (α1 , . . . , αk ) is the matrix of parameters αi . Consider for example an arbitrary hexahedron, where UP is a 3 × 3 matrix. To simplify control of the spectral properties of the matrix UP , as needed by the stability condition, we write it in a form inspired by the formulas (4.6) and (4.7):   α1 + α4 + α5 −α4 −α5   . (6.1) UP =  −α4 α2 + α4 + α6 −α6   −α5 −α6 α3 + α5 + α6 It is obvious that UP is positive definite when all parameters are non-negative and α1 , α2 and α3 are strictly positive. Let us define the function F (α1 , . . . , αk ) that measures the overall size of the off-diagonal entries in the matrix WP : X F (α1 , . . . , αk ) = |wij |2 i6=j

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Thus, the goal of the m-adaptation procedure is to solve a constrained minimization problem min F (α1 , . . . , αk ) αi

(0)

subject to λmin (UP ) ≥ Cλ+ min (WP ),

(6.2)

where λmin and λ+ min denote the smallest and the smallest positive eigenvalues of the matrix, respectively, and C is an appropriately selected scaling constant defined by the stability condition. By design of the parameter matrix (6.1), the entries of the mass matrix WP are linear polynomials of αi . Hence, F is a quadratic function of αi which is guarantied to have a minimum which is unique under additional assumptions. The conditions on the extremum of the functions F are easy to analyze, as they form a linear system of equations ∂F = 0, ∂αi

i = 1, 2, . . . , k.

(6.3)

Remark 6.1. If the family of mimetic schemes contains a diagonal matrix it will be found as a solution of the constrained minimization problem (6.3). Therefore, we recommend the minimization procedure (6.3) for meshes consisting predominantly of Voronoi cells. Remark 6.2. A simpler, but potentially suboptimal, strategy is to replace the constraints on the eigenvalues of the parameter matrix UP in (6.2) by constrains on the values of the parameters: (0)

αi ≥ Cλ+ min (WP )

for

i = 1, 2, 3

and αj ≥ 0

for j = 4, 5, 6,

which guarantee positivity of the parameter matrix UP . 6.2. Impact of m-adaptation on efficiency of AMG method In this section we perform a numerical experiment to test the improved diagonal dominance, given by (6.3), on the performance of the AMG method. We consider the elliptic equation − div K(∇p − ρg) = 0, with a low-order term ρg representing density times gravity. A similar equation is used as a simple model of pressure distribution in a subsurface water or oil reservoir. A simplified model, shown in Fig. 11, illustrates the complex configuration of geological layers (materials with different anisotropic permeabilities) and the highly distorted computational mesh. We impose hydrostatic boundary condition on the most right vertical side, the constant pressure boundary condition on the bottom side, constant infiltration rate on the top side, and no-flow boundary conditions on the remaining lateral sides. In a dynamic simulation (a term c ∂p ∂t is added to the equation), we can avoid multiple computation of the mass matrices WP by doing it only once for the whole simulation. Thus, the cost of solving the optimization problems during the initialization phase become negligent compared to the total computation time. In the numerical analysis, we are focused on the complexity of one V-cycle 111 of the particular AMG and the cost of one preconditioned conjugate gradient (PCG) solver. Even a modest reduction of the computational cost (10%-20%) is sufficient to recommend the optimization strategy for simulations that may require 103 − 105 applications of the V-cycles. As a black-box AMG preconditioner, we selected the BoomerAMG from the Hypre package 62 with a typically used set of input parameters.

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Fig. 11. A coarse hexahedral mesh and materials with different permeabilities. The domain is scaled 7 times in the vertical direction.

The Table 1 presents two numbers, labeled the ‘operator’ and the ‘grid’, describing a relative computational complexity of the V-cycle in the default, see formula (2.28), and the optimized mimetic schemes. The first number is the accumulated cost of matrix-vector multiplications on all levels of the multigrid method divided by the cost of the matrix-vector multiplication on the first level. The second number is the total number of unknowns on all levels divided by the number of original unknowns. Smaller numbers represent a more efficient coarsening strategy. The finest mesh has the highest shape-regularity properties (measured as the average cell aspect ratio, Q, in the table) which leads to the smallest computational benefits. Note that the coarser meshes are used at various stages of the UQ analysis and, hence, are also of great interest.

# cells 1910 7330 22550 108720 413010

Q 92.1 50.2 34.3 16.1 9.27

Default MFD operator 3.13 3.33 3.20 2.87 2.77

method grid 1.84 1.79 1.72 1.60 1.53

Optimized MFD method operator grid 2.32 1.56 2.34 1.52 2.29 1.48 2.29 1.48 2.40 1.42

Table 1. Complexity of V-cycle in a fully saturated elliptic solver.

An efficient multigrid coarsening does not necessarily imply a smaller solution time. In Table 2 we present CPU times (in seconds) for generating and assembling stiffness matrices and the iterative solution of the corresponding linear systems. We employed a PCG method with the relative tolerance 10−12 . Note that the m-adaptation requires roughly three times more time for generating and assembling the stiffness matrix. But the CPU time of the PCG solver is 10% - 30% smaller. Hence, less than ten calls of the iterative solver could be sufficient to offset the cost of the optimization procedure.

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M-adaptation

# cells 1910 7330 22550 108720 413010

Default MFD method matrix time PCG time 0.05 0.09 0.18 0.35 0.51 1.12 2.40 5.02 8.69 21.5

33

Optimized MFD method matrix time PCG time 0.14 0.07 0.49 0.26 1.54 0.84 6.48 4.25 24.7 19.2

Table 2. CPU times of matrix generation and problem solution.

7. Conclusion We reviewed the design principles involved in the construction of the family of MFD schemes that include the consistency and the stability conditions. By construction, the MFD family contains schemes with equivalent base properties such as the stencil size, stability region, and convergence order. Amongst the family there are members that in addition to the base properties posses additional (superior) properties compared to the rest of the schemes in the family. The strategy for identifying schemes with such properties is dubbed m-adaptation. These superior properties can be different for different applications. We presented a few examples of such properties and the m-adaption for them. For the diffusion equation the superior property is the preservation of the maximum principle for the solution in the discrete form. We showed how to perform the m-adaption for the diffusion in the primal and the dual form on some shapes of the elements to guaranty the discrete maximum principle. For the acoustic wave equation the superior property is a smaller numerical dispersion. We showed that the numerical dispersion can be reduced by two orders of the wave resolution parameter κh compared to the base order (from second to fourth) on rectangular and cuboid meshes. For the multi-grid solvers the superior property is the speed of convergence. We presented a simple m-adaptation procedure that improves the performance of the multi-grid solvers by 10% − 30% for a practical problem of subsurface flows. Acknowledgments This work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC5206NA25396. The authors acknowledge support of the DOE Office of Science Advanced Scientific Computing Research (ASCR) Program in Applied Mathematics Research. The model in Sec. 6 was provided by the DOE Office of Environmental Management Advanced Simulation Capability for Environmental Management (ASCEM) Program128 . References 1. I. Aavatsmark, T. Barkve, Ø. Bøe, and T. Mannseth. Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods. SIAM J. Sci. Comput., 19(5):1700–1716, 1998. 2. I. Aavatsmark, T. Barkve, Ø. Bøe, and T. Mannseth. Discretization on unstructured grids for inhomo-

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3.

4. 5. 6.

7. 8. 9. 10.

11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

23. 24.

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