Hybrid Difference Methods for PDEs

Hybrid Difference Methods for PDEs Youngmok Jeon

∗

October 25, 2014

Abstract We propose the hybrid difference methods for partial differential equations (PDEs). The hybrid difference method is composed of two types of approximations: one is the finite difference approximation of PDEs within cells (cell FD) and the other is the interface finite difference (interface FD) on edges of cells. The interface finite difference is obtained from continuity of some physical quantities. The main advantages of this new approach are that the method can applied to non-uniform grids, retaining the optimal order of convergence and stability of the numerical method for the Stokes equations is obtained without introducing staggered grids.

AMS(MOS) subject classification: 65N30, 65N38, 65N50 Key words: Finite difference, hybrid

1

Introduction

The finite difference method (FDM) is simple to implement and it can solves many physical problems efficiently [1, 3, 9, 8]. A main drawback of the FDM is that the method is not easy to handle problems with a complicated geometry compared to the finite element method even though this drawback can be overcome somehow by introducing proper geometric transforms [3, 10]. Another drawback is that a uniform griding is necessary to have an optimal order of convergence with a compact stencil. The finite difference approaches for Stokes and Navier-Stokes equations with the co-located grids pose check-board instability in pressure approximation, therefore, grid points for velocity fields and pressure should be located in a staggered manner to have a stable finite difference scheme, which is called as the staggered grid method [1, 3]. In this paper we introduce a new family of finite difference methods. We call these methods as the hybrid difference methods (HDM). Hybrid is meant by that the stiffness system is composed of two types of FD approximations, the cell finite difference and the interface finite difference that is based on the flux continuity on intercell boundaries. There exist already many different versions of hybridized discontinuous Galerkin methods (HDG) [2, 4, 5, 7]. The hybridized numerical schemes are composed of two steps. Firstly, localize PDEs on each cell, assuming the trace value of a solution on each cell boundary as the local Dirichlet data (the local problem solver). Secondly, set up the global stiffness ∗

Department of Mathematics, Ajou University, Suwon, Korea 305-701. This author was supported by NRF 2010-0021683. email:[email protected]

1

2

Y. Jeon

system by using the flux continuity on intercell boundaries. As a result the unknowns in the global discrete system are the assumed trace data of the solution on the skeleton of a mesh (the union of cell boundaries). The HDM is obtained by applying the above hybridization ideas to the finite difference situations. The cell finite difference approximation at interior nodes within cells corresponds to the local problem solver in the HDG, and the interface finite difference at nodes on the skeleton corresponds to the normal flux continuity on intercell boundaries. For the Stokes problem the interface finite difference is obtained from continuity of the total normal strain. By the nature of our approach the HDM can be applied to arbitrary non-uniform axis parallel grids, still retaining the optimal order of convergence as for the standard FDM on the uniform grid. Therefore, adaptive grids can be used to retain an optimal order of convergence even for singular solution problems. In the HDM the Stokes problem can be solved efficiently without needing staggered grids, which may reduce programming efforts. Moreover, boundary condition can be imposed more simply and efficiently compared to the staggered grid methods. Let us compare the HDM and HDG. The HDG is more flexible in mesh generation, therefore, adaptive mesh generation is handier in the HDG than the HDM. However, the HDM is not involved in ’variational crimes’ at all. First, the method does not include any numerical integration. Second, the boundary condition can be imposed exactly even for a curved boundary problem by using curved-boundary cells of which boundaries match the given boundary [6]. In this paper we aim at introducing new finite difference type methods. Therefore, our major efforts are given to description of the methods and computation. Concerning numerical analysis will be minimized and rigorous numerical analysis will a subject of future research. The paper is organized as follows. In section 2 we derive the hybrid difference methods for the elliptic and Stokes problems. For the Stokes problem a higher order HDM is proposed with a detailed description of static condensation. For the elliptic problem the mass conservation property and ellipticity of the HDM is proved. Error analysis of the HDM is out of the scope of this paper. In section 3 we present numerical experiments for those methods developed in Section 2. Three different grids, the uniform, graded and random grids are considered for numerical experiments. For elliptic problems we consider smooth solution and singular solution problems. In particular, it is shown that the optimal order of convergence can be recovered for the singular solution problem if one consider graded grids with a proper order. For the Stokes problem we consider a smooth solution case and the lid-driven cavity problem. It is observed that the higher order method outperforms for both a smooth solution case and the lid-driven cavity problem.

2

Hybrid Finite Difference Methods

The domain is composed of rectangular cells so that Ω = ∪N i=0 Ri . Suppose that each rectangle Ri contains 5-nodal points as in Fig. 1, the cell centered point is called the interior node and the edge centered points as the skeleton nodes. The collection of cells is denoted by Th , where h denotes the maximum diameter of cells. In this section we begin with proposing the HDM methods, which is based on the 5-point stencil finite difference formula for the elliptic and Stokes problems. Later in this section we discuss a higher order HDM for Stokes equations.

3

HDM for PDEs

p6 b

q2

b

b

R2

b

b

b

p5 u p2 b

u R0

q0 b

(u, p)

p3 u

q1 b b

p4 b

R1

u b

b

p1 Figure 1: A reference subdomain: |R0 | = h1 × k1 , |R1 | = h2 × k1 , |R2 | = h1 × k2

2.1

The Poisson problem

Let us consider a Poisson problem with the Dirichlet boundary condition: −∆u = f

on Ω,

u = 0 on Γ, where Γ = ∂Ω. Then, the solution u satisfies the cell problem −∆u = f

on R

(2.1)

for each R and it satisfies the interface condition, [[∂ν u]] = ∂ν u|e + ∂ν 0 u|e = 0 on e = ∂R ∩ ∂R0 .

(2.2)

Here, ∂ν u := ∂u ∂ν . For approximation of the cell problem (2.1) we consider the 5-point stencil FD approximation at the cell centered point q0 . Therefore, −∆h u(q0 ) :=

−u(p2 ) + 2u(q0 ) − u(p3 ) −u(p5 ) + 2u(q0 ) − u(p1 ) + = f (q0 ). (h1 /2)2 (k1 /2)2

(2.3)

It is well known that −∆h u = −∆u + O(h2 ). At the interface p3 , the interface condition becomes [[ux ]] = 0. By the well known 2nd order upwind finite difference formula: ux (p3 ) =

3u(p3 ) − 4u(q0 ) + u(p2 ) + O(h21 ), h1

ux (p3 ) = −

3u(p3 ) − 4u(q1 ) + u(p4 ) + O(h22 ) h2

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Y. Jeon

from R0 and R1 , respectively. Equating the above two equations, we obtain the finite difference approximation of (2.2) at p3 as follows; [[∂νh u]]p3 =

3u(p3 ) − 4u(q0 ) + u(p2 ) 3u(p3 ) − 4u(q1 ) + u(p4 ) + = 0. h1 h2

(2.4)

Similarly, we will have at p5 [[∂νh u]]p5 =

3u(p5 ) − 4u(q0 ) + u(p1 ) 3u(p5 ) − 4u(q2 ) + u(p6 ) + = 0. k1 k2

(2.5)

Equation (2.3) will be called the cell finite difference (cell FD) and Equations (2.4) and (2.5) are called the interface finite difference (interface FD), respectively. Combining (2.3), (2.4) and (2.5) we obtain the HDM method in a symbolic form: −∆h u(q) = f (q),

[[∂νh u]]p = 0

(2.6)

for each interior node q and skeleton node p. Let us introduce an equivalence class for the continuous function space with vanishing trace, C0 (Ω). Two functions, u, v ∈ C0 (Ω) are equivalent (u ≡ v) if u = v at all the cellcenter and skeleton nodes. The equivalence class is denoted by C0h (Ω). For u, v ∈ C0h we introduce two discrete inner products: (u, v)h =

X

(u, v)R,h ,

hu, vih =

R∈Th

X

hu, vi∂R,h ,

R∈Th

P where (u, v)R,h = |R|u(q)v(q) and hu, vi∂R,h = e⊂∂R |e|u(pe )v(pe ) with q the cell center of a cell R and pe the edge center of an edge e. The induced norm is defined by kuk2h = (u, u)h . The equation (2.6) can be rewritten in a variational form; find u ∈ C0h (Ω) such that −(∆h u, v)h + h∂νh u, vih = (f, v)h ,

v ∈ C0h (Ω).

(2.7)

Remark 2.1. In the HDM the essential unknowns are only u(p)’s on the skeleton nodes since u(q) on the interior nodes can be solved in terms of u(p)’s from the cell FD. In view of the reference cell R0 in Fig. 1 let us introduce cell centered finite differences: u(p3 ) − u(p2 ) , h1 4u(p3 ) − 8u(q0 ) + 4u(p2 ) h Dxx u(q0 ) = , h21 Dxh u(q0 ) =

u(p5 ) − u(p1 ) , k1 4u(p5 ) − 8u(q0 ) + 4u(p1 ) h Dyy u(q0 ) = , k12 Dyh u(q0 ) =

R 1 Theorem 2.2. If one replace f in (2.3) with f = |R| R f dx, the HDM satisfies the mass conservation, that is, Z Z h f (x)dx. ∂ν uds = − ∂R0

Here,

∂νh u

R0

is assumed to be constant on each edge.

5

HDM for PDEs

Proof. Simple calculation yields Z 3u(p3 ) − 4u(q0 ) + u(p2 ) 3u(p5 ) − 4u(q0 ) + u(p1 ) ∂νh uds = k1 + h1 h1 k1 ∂R0 3u(p1 ) − 4u(q0 ) + u(p5 ) 3u(p2 ) − 4u(q0 ) + u(p3 ) k1 + h1 + h1 k1 Z f (x)dx = (Dxx u(q0 ) + Dyy (q0 )h1 k1 = − R0

Theorem 2.3. For u, v ∈ C(R0 ), −(∆h u, v)R0 ,h + h∂νh u, vi∂R0 ,h =

1 1 h h h h (h1 Dxx u, h1 Dxx v)R0 ,h + (k1 Dyy u, k1 Dyy v)R0 ,h 8 8 +(Dxh u, Dxh v)R0 ,h + (Dyh u, Dyh v)R0 ,h

on the reference cell R0 in Fig. 1. Proof. Simple calculation yields that −(∆h u, v)R0 ,h + h∂νh u, vi∂R0 ,h = S1 + S2. Now, S1 = −Dxx u(q0 )v(q0 )h1 k1 + ∂νh u(p3 )v(p3 )k1 + ∂νh u(p2 )v(p2 )k1 k1 k1 = (−4u(p2 ) + 8u(q0 ) − 4u(p3 ))v(q0 ) + (3u(p2 ) − 4u(q0 ) + u(p3 ))v(p2 ) h1 h1 k1 +(3u(p3 ) − 4u(q0 ) + u(p2 ))v(p3 ) h1 k1 {(2u(p2 ) − 4u(q0 ) + 2u(p3 ))(−2v(q0 ) + v(p2 ) + v(p3 )) + (u(p2 ) − u(p3 ))(v(p2 ) − v(p3 ))} = h1 1 2 = h (Dxx u, Dxx v)R0 ,h + (Dx u, Dx v)R0 ,h . 8 1 By a similar way, S2 = −Dyy u(q0 )v(q0 )h1 k1 + ∂νh u(p5 )v(p5 )h1 + ∂νh u(p1 )v(p1 )h1 1 2 = k (Dyy u, Dyy v)R0 ,h + (Dy u, Dy v)R0 ,h . 8 1 The theorem is proved. The following ellipticity is immediate from the above theorem. Therefore, the HDM (2.7) is uniquely solvable on C0h (Ω). Corollary 2.4. For u ∈ C0 (Ω) and on a quasi-uniform grid, h h −(∆h u, u)h + h∂νh u, uih ≥ Ch(kDxx uk2h + kDyy uk2h ) + kDxh uk2h + kDyh uk2h

with some constant C > 0.

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Y. Jeon

2.2

The Stokes problem

Let us consider a Stokes problem: −∆u + ∇p = f

on Ω,

divu = 0

on Ω,

u = 0 with

R

Ωp

on Γ

dx = 0. Then, the solution of the above Stokes problem satisfies the cell problem, −∆u + ∇p = f

on R,

(2.8a)

divu = 0

on R,

(2.8b)

on e = ∂R ∩ ∂R0

(2.9)

and the interface condition [[−∂ν u + pν]] = 0

R with Ω p dx = 0. To obtain a stable FD approximation of the Stokes problem we need the following considerations. L1: The degrees of freedom for the pressure are located only on the interior nodes {qi }. Therefore, for the FD approximation of ∇p we use only the pressure at the interior nodes within a cell. L2: The degrees of freedom for the velocity field are located at both the interior and skeleton nodes, {qi }∪{pj }. Therefore, ∆u and divu are approximated by using both the interior and skeleton nodes in a cell. L3: To approximate the pressure on the skeleton(which appears in the interface condition) the extrapolation pE of the interior pressure p is used. As a result, pE is assumed to be a piecewise function on Ω. Since we have only one interior node in a cell for the five point FD method we consider that the pressure is constant in each cell, say, pE = p(qi ) and ∇h pE = 0 on each Ri . Hence, the FD approximations for the equation (2.8a) and (2.8b) become −u(p2 ) + 2uh (q0 ) − u(p3 ) −u(p5 ) + 2uh (q0 ) − u(p1 ) + (h1 /2)2 (k1 /2)2 = f (q0 ) u1 (p3 ) − u1 (p2 ) u2 (p5 ) − u2 (p1 ) + =0 divh (q0 ) = h1 k1

−∆h u(q0 ) =

(2.10a)

(2.10b)

on the cell R0 with u = (u1 , u2 ). The interface condition (2.9) yields the interface FD:  3u1 (p3 )−4u1 (q0 )+u1 (p2 )  + 3u1 (p3 )−4uh12(q1 )+u1 (p4 ) − pE (q0 ) + pE (q1 ) = 0  h1 [[∂νh u − pE ]]p3 =  3u2 (p3 )−4u2 (q0 )+u2 (p2 )  + 3u2 (p3 )−4uh22(q1 )+u2 (p4 ) = 0 h1 (2.11)

7

HDM for PDEs

and [[∂νh u

E

− p ]]p5 =

    

3u1 (p5 )−4u1 (q0 )+u1 (p1 ) k1

+

3u1 (p5 )−4u1 (q2 )+u1 (p6 ) k2

=0 .

3u2 (p5 )−4u2 (q0 )+u2 (p1 ) k1

+

3u2 (p5 )−4u2 (q2 )+u2 (p6 ) k2

−

pE (q

0)

+

pE (q

2)

=0 (2.12) Combining (2.10), (2.11) and (2.12) we have the HDM for the Stokes problem in a symbolic form: −∆h u = f

on R,

(2.13a)

divh u = 0

on R,

(2.13b)

and and

R

[[∂νh u − pE ν]] = 0 on p ∈ ∂R ∩ ∂R0 Ωp

E

(2.14)

dx = 0. The equations (2.13) and (2.14) can be rewritten in a variational form: −(∆h u, v)h + h∂νh u − pE ν, vih = (f , v)h ,

v ∈ [C0 (Ω)]2

(divh u, q)R,h = 0 R ∈ Th , q ∈ R.

(2.15a) (2.15b)

By Theorem 2.3 we will have −(∆h u, v)h + h∂νh u, vih =

1 X h h h h (hR Dxx u, hR Dxx v)R,h + (kR Dyy u, kR Dyy v)R,h 8 R∈Th

+(Dxh u, Dxh v)h + (Dyh u, Dyh v)h . Simple calculation yields that hpE ν, viR0 ,h = p(q0 )(v1 (p2 ) − v1 (p3 ))k1 + p(q0 )(v2 (p5 ) − v2 (p1 ))k1 = (p, divh v)R0 ,h for v = (v1 , v2 ). Let us introduce bilinear forms: ah (u, v) = −(∆h u, v)h + h∂νh u, vih ,

bh (p, u) = (p, divh u)h .

Then, the equation (2.15) can be rewritten as follows: find (u, p) ∈ [C0h (Ω)]2 × P0 (Ω) such that ah (u, v) + bh (p, v) = (f, v), b∗h (u, q)

= 0,

v ∈ [C0h (Ω)]2

q ∈ P0 (Ω),

(2.16a) (2.16b)

where b∗h (u, q) = bh (q, u) and P0 (Ω) ⊂ L2 (Ω) is the space of piecewise constant functions. R Theorem 2.5. Suppose P ∈ P0 (Ω) is a piecewise function on a mesh Th with Ω P dx = 0. Then, there exists a u ∈ [C0 (Ω)]2 such that divh u = P . Proof. We provide a sketch of the proof. Consider a simplest subdivison of Ω that is composed of four rectangles as in Fig. 2. The boundary values are u = (0, 0) and we assume u(p9 ) = (α0 , 0), u(p10 ) = (0, α1 ), u(p11 ) = (α2 , 0) and u(p12 ) = (0, α3 ). We will prove that there

8

Y. Jeon

p6

p5 b

b

p7

b

q3

b

b

b

b

q0 b

b

p4 b

b

p10

p12 p8

p11 q2

p9

b

p1

b

q1 b

b

p3

b

p2

Figure 2: R0 = h1 × k1 , R1 = h2 × k1 , R2 = h2 × k2 and R3 = h1 × k2

exists a solution (α0 , α1 , α2 , α3 ) that satisfies divh u(qi ) = Pi (i = 0, 1, 2, 3) if P0 h1 k1 + P1 h2 k1 + P2 h2 k2 + P3 h1 k2 = 0. From divh u(qi ) = Pi we obtain the following four equations. α0 k1 + α3 h1 = P0 h1 k1 , −α1 k2 − α2 h2 = P2 h2 k2 ,

R

Ω

P dx =

−α0 k1 + α1 h2 = P1 h2 k1 , α2 k2 − α3 h1 = P3 h1 k2 .

R The above equation is solvable for (α0 k1, α1 h2 , α2 h2 , α3 h1 ) if Ω P dx = 0. This idea can be extended to prove the theorem on a finer mesh. Suppose Ω is a square and it is divided into a N × N subdivisons. There exist 2N (N − 1) interior edges. The equation divh (u) = P produces a linear system consisting of N 2 equations 2 with 2N (N R − 1) degrees of freedom. The rank of this linear system is (N − 1) and the condition Ω P dx = 0 provides a compatibility condition for this linear system. Theorem 2.5 implies that the inf-sup condition is satisfied by the HDM for the Stokes problem. Z bh (p, u) sup ≥ kpkh , p ∈ P0 (Ω), pdx = 0. Ω u∈[C h (Ω)]2 kdivh ukh 0

Therefore, ellipticity of ah and the inf-sup condition implies that the HDM is uniquely solvable. Remark 2.6. The 5-point finite difference method can be comparable with the N Q1 × P0 finite element method for the Stokes problem. Here, N Q1 is the non-conforming, rotated Q1 -element.

2.3

A higher order method for the Stokes Problem

Let Ω = ∪N i=0 Ri and each cell contains 9-interior nodes and 12-skeleton nodes as depicted in Fig. 3. Let us denote the set of all skeleton nodes interior of Ω by SΩ . The HDM is constructed in the following manner:

9

HDM for PDEs

p7

p5

b

b

b

p8 p9

p6

b

b

q7 b

b

q1 b

b b

q4 b

p4

b

q8

q9

q5

q6

q2

q3

b

b

b

b

b

b

b

b

b

p10

p11

p12

p3 p2 p1

R Figure 3: A reference cell: |R| = h × k

H1: All the nodes are the Gaussian points. 9 H2: The pressure pE 2 is a quadratic polynomial which is based on the interior nodes {qi }i=1 . For example, on the line with nodes {p8 , q4 , q5 , q6 , p2 }

pE 2 (x) = p(q4 )

(x − q5 )(x − q6 ) (x − q4 )(x − q6 ) (x − q4 )(x − q5 ) + p(q5 ) + p(q6 ) (q4 − q5 )(q4 − q6 ) (q5 − q4 )(q5 − q6 ) (q6 − q4 )(q6 − q5 )

H3: By a similar way the velocity field u4 is an interpolation polynomial of degree four, based on both the interior and skeleton nodes, {qi }9i=1 ∪ {pk }12 k=1 . E Set ∆u4 = ∆h u, divu4 = divh u , ∂ν u4 = ∂νh u and ∇pE 2 = ∇h p . Then, the cell FD becomes

−∆h u(qi ) + ∇h pE (qi ) = f (qi ), divh u(qi ) = 0,

i = 1, · · · , 9,

i = 1, · · · , 9,

(2.17a) (2.17b)

for each R, and the interface FD is [[−∂νh u + pE ν]]p = 0,

p ∈ SΩ .

(2.18)

For a higher order method with many interior nodes it is desirable to use the static condensation property to reduce degrees of freedom. To induce a condensed finite difference formula E + p with a piecewise constant function p . let us set pE = pf c c E (q ))}9 The equations (2.17) and (2.18) can be reduced as follows: Solve for {(u(qi ), pf i i=1 9 12 in terms of {u(pk )}k=1 and {f (qi )}i=1 the cell finite difference approximation, E (q ) = f (q ), −∆h u(qi ) + ∇h pf i i

divh u(qi ) + KR = 0, Z E dx = 0 pf

i = 1, · · · , 9,

i = 1, · · · , 9,

KR ∈ R,

(2.19a) (2.19b) (2.19c)

R

for each cell. Then, the interface FD and divergence free condition change to E ν + p ν]] = 0, [[−∂νh u + pf c p

p ∈ SΩ ,

kR = 0 for each cell R,

(2.20a) (2.20b)

10

Y. Jeon

u p

before 12(N − 1)N + 18N 2 9N 2

after 12(N − 1)N N2

Table 1: Reduction in degrees of freedom by static condensation which yields a square system with unknowns {u(p)}p∈SΩ and pc . Remark 2.7. Table 1 shows reduction in degrees of freedom before and after the static condensation for a N × N partition of a square domain. Remark 2.8. The local problem (2.19) can be understood as a process to produce PDEE (q ))}9 adapted basis {(u(qi ), pf i i=1 for each basis of the space of cell boundary data, Vu = 2 12 {(u(pk ))k=1 : u ∈ [C(R)] }. Those basis of Vu do not naturally satisfy the divergence free condition, however, we can require local solutions to satisfy the divergence condition up to a constant. Actually, KR0 is determined by the cell boundary data {u(pi )}12 =1 as follows. 12

KR0 = −

1 X wi u(pi ) · ν(pi ) |R| i=1

with the edgewise Gaussian weight {wi }12 i=1 . Then, KR = 0 is imposed in (2.20) to have numerical solutions that satisfy the divergence free condition exactly.

Remark 2.9. The following approximation property is well-known in the theory of the finite difference approximation. −∆h u = −∆u + O(h3 ),

∇h u = ∇u + O(h4 ),

∇h p = ∇p + O(h2 ),

3 pE 2 = p + O(h ).

In view of the theory of FDM one might expect the error estimate ku − uh k0,h = O(h3 ), where k · k0,h is the discrete L2 -norm. According to our numerical experiments in §3 a superb convergence ku − uh k0,h = O(h4 ) is obtained. Moreover, we also have a superb convergence result for p, kp − ph k0,h = O(h3.5 ). Apparently, it looks kp − ph k0,h . kp − pE 2 k0,h . Here, (uh , ph ) is the HDM approximation of a smooth exact solution (u, p). Remark 2.10. For the numerical experiments in §3 we use the Gaussian nodes for the HDM as claimed in H1. However, we observe experimentally that the same order of convergence is maintained even with non-Gaussian nodes. The q nodes in Fig. 3 is based on qGaussian the tensor of one dimensional Gaussian nodes, {− √1 3

3 5 , 0,

3 5 }.

One may use non-Gaussian

≤ q < 1, which is the condition to have positive weights in the nodes, {−q, 0, q} with corresponding quadrature formula.

11

HDM for PDEs

A graded grid

A random grid

A uniform grid

1

1

1

0.9

0.9

0.9

0.8

0.8

0.8

0.7

0.7

0.7

0.6

0.6

0.6

0.5

0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0

0

0

0.5

1

0

0.5

0

1

0

0.5

1

Figure 4: The graded, random and uniform grids

3

Numerical experiments

We consider three different grids: the graded, uniform and random grids on the domain [0, 1] × [0, 1] as shown in Fig. 4. The graded mesh used in numerical experiments is of order 2, which is based on the mesh grading transformation, (˜ x, y˜) = (

y2 x2 , ), x2 + (1 − x)2 y 2 + (1 − y)2

where (x, y) is a uniform griding. Hence, at four corners of the domain, we have finer grids as in an adaptive mesh refinement.

3.1

The Poisson problem

Let us consider two Poisson problems; one with a smooth solution and the other with a singular solution.

L

2

−2

error

L

2

−2

10

error

10 Graded Random Uniform

Graded Random Uniform

−3

−3

10

error

error

10

−4

10

−5

−4

10

−5

10

10 h2

h2

−6

10

−6

0

10

1

10 N

2

10

10

0

10

1

10 N

2

10

Figure 5: The L2 -convergence of numerical solutions for Ex. 3.1 (left) and Ex. 3.2 (right).

12

Y. Jeon

Ex 3.1. −∆u = exp(x1 ) u = g

in Ω,

on Γ,

where the Dirichlet data g is given to have the exact solution u(x1 , x2 ) = exp(x1 )(sin(x2 )+ 1). Ex 3.2. −∆u = 0

in Ω,

u = g

on Γ,

where g is given to have the exact solution u(x1 , x2 ) = real part of (x1 + ix2 )1/2 For the smooth solution case (Ex. 3.1) the graded and uniform grids yield numerical solutions with the same order of convergence, O(h2 ). It is worth to note that the non-smooth solution in Ex. 3.2 has a regularity, u ∈ H 3/2− (Ω) for arbitrary small  > 0. Hence, on a uniform grid the possible order of convergence is of O(h3/2− ). By using a graded mesh the optimal order of convergence, O(h2 ) can be restored (see Fig. 5). The random grid yields an erratic convergence result by its own nature, however, the HDM works quite well even in this case. Table 2 compares the condition numbers of the uniform and graded grids for Ex. 1. N 4 8 16 32 64

Ch 5.5771 e+2 3.2690 e+3 2.1328 e+4 1.5157 e+5 1.1371 e+6

α 2.5512 2.7059 2.8291 2.9073

Ch 2.4157 e+3 1.1085 e+5 4.4466 e+6 1.5882 e+8 5.3556 e+9

α 5.5199 5.3261 5.1586 5.0756

Table 2: Comparison of condition numbers for the uniform and graded meshes. log(Ch /C2h )/ log(2).

α =

Since the graded grid is not quasi-uniform the condition number increases in a much higher order, O(h−5 ) than that of the uniform grid, which implies that the graded grid method is less stable.

3.2

The Stokes problem

We consider a smooth solution problem to test convergence property for both the low order method (that in §2.2) and the higher order method (that in §2.3). We also test performance of our methods with a well-known benchmark problem, the lid-driven cavity problem. Here, we consider only two grids, the graded and uniform grids.

13

HDM for PDEs

Ex 3.3. Consider a Stokes problem,

Z



 2x1 on Ω, 0 divu = 0, on Ω

−∆u + ∇p =

pdx = 0, Ω

where the exact solution is (u1 , u2 ) = (exp(x1 ) sin(x2 ), exp(x1 ) cos(x2 )) and p = x21 − 1/3. N 4 8 12 16 20 24

ku − uh k0,h 7.7915 e-6 5.0101 e-7 9.9837 e-8 3.1724 e-8 1.3027 e-8 6.2926 e-9

d.o.f 160 736 1728 3136 4960 7200

α 3.9590 3.9783 3.9852 3.9888 3.9909

kp − ph k0,h 3.8992 e-5 3.4945 e-6 8.4936 e-7 3.1105 e-7 1.4264 e-7 7.5427 e-8

α 3.4800 3.4885 3.4918 3.4937 3.4948

Table 3: The convergence results for the higher order method in Section 2.3. The term, d.o.f means degrees of freedom.

With the lower order HDM it is expected O(h2 )-convergence in approximation u and O(h)-convergence in p for both grids if the solution (u, p) is regular enough. We observe the expected convergence for u and a higher order of convergence O(h1.5 ) for p (Fig. 6). For a higher order method, we observe O(h4 ) convergence for u and O(h3.5 ) for p (see Table 3). L

2

−1

error in u

L

2

−1

10

error in p

10 Graded Uniform

Graded Uniform

−2

error

error

10

−2

10

−3

10

1.5

h

2

h

−4

10

−3

0

10

1

10 N

2

10

10

0

10

1

10 N

2

10

Figure 6: The L2 -convergence results for the velocity field (left) and the pressure (right)

14

Y. Jeon Streamline of (u , u ) 1

Secondary vortex

2

0.1 1 0.09

0.9

0.08

0.8 0.7

0.07

0.6

0.06

0.5

0.05

0.4

0.04

0.3

0.03

0.2

0.02

0.1

0.01

0 0

0.2

0.4

0.6

0.8

1

0 0.88

0.9

0.92

0.94

0.96

0.98

1

Figure 7: Streamline of a lid-driven cavity flow(left) and a closeup of the secondary flow(right) on the 32 × 32-uniform grid. Streamline of (u , u ) 1

Secondary vortex

2

0.09 1 0.08

0.9 0.8

0.07

0.7

0.06

0.6

0.05

0.5 0.04

0.4 0.3

0.03

0.2

0.02

0.1

0.01

0 0

0.2

0.4

0.6

0.8

1

0 0.9

0.92

0.94

0.96

0.98

1

Figure 8: Streamline of a lid-driven cavity flow(left) and a closeup of the secondary flow(right) on the 32 × 32-graded grid. In Table 3, the pair (uh , ph ) is the HDM numerical solutions for the exact solutions, (u, p), and α represents the rate of convergence. The discrete L2 -norm, k · k0,h is computed by the composite Gaussian quadrature. In view of numerical results and Remark 2.9 rigorous convergence analysis must be an interesting subject of future research. Ex 3.4. Consider the driven-cavity flow problem −∆u + ∇p = 0

in Ω

∇·u = 0

in Ω

with the watertight boundary condition:  (1, 0)T , 0 < x1 < 1, x2 = 1, u(x1 , x2 ) = (0, 0)T , otherwise. For the lid-driven cavity problem the exact solution is not regular enough to guarantee the optimal order of convergence for the uniform mesh. The graded mesh should yield a more accurate approximation for (u, p) in view of the secondary flows in Figs. 7 and 8. Fig. 9 is

15

HDM for PDEs Streamline of (u , u ) 1

Secondary vortex

2

0.1 1 0.09

0.9

0.08

0.8 0.7

0.07

0.6

0.06

0.5

0.05

0.4

0.04

0.3

0.03

0.2

0.02

0.1

0.01

0 0

0.2

0.4

0.6

0.8

1

0 0.9

0.92

0.94

0.96

0.98

1

Figure 9: Streamline of a lid-driven cavity flow(left) and a closeup of the secondary flow(right) on the 20 × 20-uniform grid with the high order method. obtained by using the higher order HDM in §2.3 on the 20 × 20 mesh with the total degrees of freedom, 4960, while Figs. 7 and 8 are obtained on the 32 × 32 mesh with the similar total degrees of freedom, 4992. Fig. 9 shows much more details in the secondary flow closeup. This is the reason why higher order methods are preferred in many flow simulations even when the exact solutions are expected to be not regular enough.

References [1] S. Armfield and R. Street, A comparison of staggered and non-staggered grid NavierStokes solutions for the 8:1 cavity natural convection flow, ANZIAM J. 46 (E), C918C934 (2005) [2] B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319–1365. [3] J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics, Springer, 2002 [4] Y. Jeon and E.-J. Park, A hybrid discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal. 48 (2010), 1968-1983 [5] Y. Jeon and E.-J. Park, New locally conservative finite element methods on a rectangular mesh, Numer. Math. 123 (2013), 97-119 [6] D. Shin, Y. Jeon and E.-J. Park, A research note-HDM. [7] Y. Jeon and D. Sheen, A locking-free locally conservative hybridized scheme for elasticity problems, Japan J. of Indust. and Appl. Math., 30 (2013) 585-603 [8] J. Kim and P. Moin, Application of a fractional-step method to incompressible NavierStokes equations, J. of Comp. Physics, 59 (1985) 308-323 [9] R. J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge Tests in Applied Math., Cambridge University Press, 2002

16

Y. Jeon

[10] C. M. Rhie and W. L. Chow, Numerical study of the turbulant flow past an airfotl with trailing edge separation, AIAA Journal, 21 (1983) 1525-1532