ON THE FINITE VOLUME ELEMENT METHOD FOR

SIAM J. NUMER. ANAL. Vol. 35, No. 5, pp. 1762–1774, October 1998

c 1998 Society for Industrial and Applied Mathematics °

004

ON THE FINITE VOLUME ELEMENT METHOD FOR GENERAL SELF-ADJOINT ELLIPTIC PROBLEMS∗ HUANG JIANGUO† AND XI SHITONG‡ Abstract. The finite volume element method (FVE) is a discretization technique for partial differential equations. This paper develops discretization energy error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations, on which there exist linear finite element spaces, and a very general type of control volumes (covolumes). The energy error estimates of this paper are also optimal but the restriction conditions for the covolumes given in [R. E. Bank and D. J. Rose, SIAM J. Numer. Anal., 24 (1987), pp. 777–787], [Z. Q. Cai, Numer. Math., 58 (1991), pp. 713–735] are removed. The authors finally provide a counterexample to show that an expected L2 -error estimate does not exist in the usual sense. It is conjectured that the optimal order of ku − uh k0,Ω should be O(h) for the general case. Key words. finite element, finite volume, discretization, error estimates AMS subject classifications. 65N10, 65N30 PII. S0036142994264699

1. Introduction. The finite volume element method (FVE) is a discretization technique for partial differential equations, especially those that arise from physical conservation laws including mass, momentum, and energy. FVE uses a volume integral formulation of the original problem and a finite partitioning set of covolumes to discretize the equations; meanwhile, the approximate solution is chosen out of a finite element space [3], [4], [5]. FVE and the finite volume method (FV) widely used in computational fluid mechanics and heat transfer [9] can both be considered as some special weighted residual methods. They possess the important and crucial property of inheriting the physical conservation laws of the original problem locally. Thus they can be expected to capture shocks, to produce simple stencils, or to study other physical phenomena more effectively. But, in contrast to FV, FVE has another remarkable characteristic, i.e., it preserves more mathematical structures of the original continuous problem. Therefore we can obtain systematic error analyses for this method in a relatively standard way. FVE also has some other virtues; see [3], [4], [5] for more details. Now let us introduce some known results about FVE. In [2], FVE (called the box method in that paper) for the model Dirichlet boundary value problem was considered; the order of the energy error estimate is just the same as that of usual conforming finite element method. In [3], accuracy estimates for FVE for diffusion equations on simple composite grids were established. However, the proofs are tedious. In [4], Z. Q. Cai, J. Mandel, and S. McCormick developed a simple theory for diffusion equations for general finite element triangulations. But it can be applied only to a special choice of control volumes, i.e., the Voronoi meshes. In [5], Z. Q. Cai presented some optimal ∗ Received by the editors March 14, 1994; accepted for publication (in revised form) June 16, 1997; published electronically June 29, 1998. This research was supported in part by the Science Foundation of Shanghai Jiaotong University for Young Teachers. http://www.siam.org/journals/sinum/35-5/26469.html † Department of Applied Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P.R. China ([email protected]). ‡ Department of Power Machinery Engineering, Shanghai Jiao Tong University, Shanghai 200030, P.R. China.

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energy error estimates for FVE (including the effects of numerical integration) for selfadjoint elliptic boundary value problems, under the conditions that the finite element triangulation and its dual are both regular, and the uniform ellipticity of the finite volume operator was satisfied. Nevertheless, for the general coefficients case Cai only proved the uniform ellipticity when the dual is Voronoi mesh and each triangle in the finite element triangulation is right or isosceles. The FVE ideas can also apply to other problems [7], [8], where S. Chondhury and R. A. Nicolaides dealt with the div-curl system and obtained some rigorous numerical analyses for the MAC method. In this paper, we shall first apply the idea of [2] to obtain an important integral identity and then combine it with the projection technique to demonstrate the uniform ellipticity of finite volume operator for general self-adjoint elliptic boundary value problems, general finite element triangulations, and general control volumes. We shall subsequently give the optimal energy estimate with more detailed analyses. In particular, the regularity assumption on the dual meshes [2], [5] is removed. Finally, we shall provide a counterexample to show that an expected L2 -estimate does not exist, which exceeds our expectation, more or less. 2. FVE and the energy estimate. Consider the general self-adjoint elliptic boundary value problem described as follows:    −∇ · (A∇u) + au = f in Ω, u = 0 on D, (2.1)   (A∇u) · n = g on N, where Ω is a plane polygonal domain with the boundary Γ divided into D and N, and meas(D) >0; f ∈ L2 (Ω), g ∈ L2 (N ), and a ∈ L∞ (Ω) (a ≥ 0 almost everywhere (a.e.) in Ω) are three given real-valued functions; A = (aij )2×2 ∈ (W 1,∞ (Ω))4 is a given real symmetric matrix-valued function. We assume that A satisfies the following ellipticity condition: there exists a constant α1 >0 such that (2.2)

α1 ξ T ξ ≤ ξ T A(x1 , x2 )ξ

¯ In what follows we shall adopt the standard definitions of ∀ξ ∈ R2 and (x1 , x2 ) ∈ Ω. Sobolev spaces and their norms and seminorms as presented in [6], and we shall also use C to denote a generic constant independent of h and other relative parameters which may be different in different places. Let {K}K∈Th be a triangulation of Ω with the diameter size h. Its nodal points 1 +M2 (the last M2 points belonging to D). Based on this trianare denoted by {zi }M i=1 gulation we construct the usual piecewise linear conforming finite element space ¯ : v|K is linear ∀K ∈ Th } S h = {v ∈ C 0 (Ω) h 1 1 (Ω) = S h (Ω) ∩ HD (Ω), where HD (Ω) = {v ∈ H 1 (Ω) : v = 0 on D}. and SD Assume that Th is regular [6], i.e., there exists a positive constant θ0 independent of h, such that, ∀K ∈ Th ,

(2.3a)

θK ≥ θ 0 ,

where θK denotes the smallest interior angle of K; or equivalently, there is a positive constant σ independent of h, such that, ∀K ∈ Th , (2.3b)

hK ≤ σ, ρK

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HUANG JIANGUO AND XI SHITONG

Fig. 2.1.

where hK = diam(K) and ρK = sup{diam(R): R is a circle contained in K}. Furthermore, to avoid unessential complexity, we assume that no interior angle of any triangle in Th is larger than π2 . In order to give the descriptions of FVE, for each interior nodal point zi we build a corresponding covolume Vi as follows: choose any interior point or median, zK , of K ∈ Th with zi as its vertex, and connect it with the medians of K [5]. Moreover, we also associate a corresponding (boundary) covolume with each boundary nodal point. The procedure is illustrated in Figure 2.1, where the covolume for boundary nodal point A is the polygonal domain PATSRQ. Thus we finally obtain a group of covolumes 1 +M2 covering the domain Ω, which is called a dual of the given triangulation {Vi }M 1 Th [7], [8]. The well-known duals are the so-called Voronoi meshes associated with the circumcenters and the Donald meshes related to the centroids. We next give some definitions about the indices of nodal points. As given in [5], let ωi = {j : zi and zj are distinct vertices of some K ∈ Th }, ω = {(i, j) : 1 ≤ i, j ≤ M1 + M2 , j ∈ ωi }, and νij = ∂Vi ∩ ∂Vj with the unit outward normal vector nij (outward with respect to Vi ); further, let zij be the line segment connecting zi and zj , and let |νij |, |zij | denote their Euclidean lengths, respectively, (i, j) ∈ ω. h (Ω) After the above preparations, the FVE for (2.1) reads as follows: find uh ∈ SD such that, for i = 1, 2, . . . , M1 , Z Z Z Z (2.4) (A∇uh ) · nds + a(x)dxuh (zi ) = f (x)dx + g(x)ds. − ∂Vi \N

Vi

Vi

∂Vi ∩N

h 1 Let {φi }M i=1 represent the conventional shape basis functions of SD (Ω) such that

uh =

M1 X

yi φi .

i=1

Then (2.4) amounts to the following linear system: (2.5)

By = [bij ]M1 ×M1 y = f¯,

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where f¯ = [f1 , f2 , . . . , fM1 ]T with

 Z  f (x)dx, Vi is an interior covolume,   Vi fi = Z Z    f (x)dx + g(x)ds, Vi is a boundary covolume, Vi

∂Vi ∩N

and y = [y1 , y2 , . . . , yM1 ]T ,  Z     − ∂V \N (A∇φj ) · nds, i 6= j, i Z Z bij =    (A∇φ ) · nds + a(x)dx, − i  ∂Vi \N

Vi

i = j.

Here T denotes the transpose operator of a matrix. In general, the coefficient matrix B is not symmetric. But there are two exceptions below. 1. A is a constant symmetric matrix. 2. A = c(x)I, zK is chosen to be the circumcenter of K. It is easy to see afterwards that B is ill conditioned. Therefore we should use PCG or GMRES combined with domain decomposition techniques to solve (2.5). From now on, for simplicity, we shall still use ∂Vi to denote ∂Vi \ N for boundary covolume Vi when there is no danger of ambiguity. Let us define W h = {v : v|K = constant,

K ∈ Th }.

We then have an important identity confirmed in the following lemma. Lemma 2.1. For arbitrary symmetric matrix-valued function A¯ ∈ (W h )4 , v ∈ h SD (Ω), −

M1 Z X i=1

∂Vi

¯ · nds v(zi ) = (Av)

Z Ω

¯ (A∇v) · ∇vdx.

Proof. Clearly it suffices to prove that for any two shape basis functions φj , φk , −

M1 Z X i=1

∂Vi

¯ j ) · nds φk (zi ) = (A∇φ

Z Ω

¯ j ) · ∇φk dx, (A∇φ

i.e., Z (2.6)

−

∂Vk

¯ j ) · nds = (A∇φ

Z Ω

¯ j ) · ∇φk dx, (A∇φ

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HUANG JIANGUO AND XI SHITONG

Fig. 2.2.

which can be proved in the succedent steps. If (j, k) ∈ ω as described in Figure 2.2, we only have to obtain the following identity which implies (2.6): Z Z ¯ ¯ j ) · ∇φk dx. (2.7) (A∇φj ) · nds = (A∇φ − A4 A5 A6

K

It follows from Green’s formula that Z Z ¯ j ) · nds = − ¯ j )dx (A∇φ ∇ · (A∇φ − A4 A5 A6 A1 A4 A5 A6 Z Z (2.8) ¯ j ) · nds + (A∇φ + A1 A4

A1 A6

¯ j ) · nds (A∇φ

¯ j ) · n|A A |A1 A6 |, ¯ j ) · n|A A |A1 A4 | + (A∇φ =(A∇φ 1 4 1 6 where n denotes the unit outward normal vector on the boundary of the quadrilateral A1 A4 A5 A6 . On the other hand, we have (2.9) Z K

¯ j ) · ∇φk dx = − (A∇φ

Z K

¯ j )φk dx ∇ · (A∇φ

Z ¯ j ) · nφk ds (Aφ + ∂K  Z Z Z ¯ j ) · nφk ds (A∇φ = + + A1 A2 A2 A3 A3 A1 Z ¯ ¯ φk ds + (A∇φj ) · n =(A∇φj ) · n A1 A2 A1 A2

A1 A3

Z A1 A3

φk ds

1 ¯ 1 ¯ |A1 A3 |. = (A∇φj ) · n A1 A2 |A1 A2 | + (A∇φj ) · n 2 2 A1 A3 Thus (2.7) follows from (2.8) and (2.9) since A4 , A6 are the medians. If j = k, the proof for (2.6) is similar to the above. For the other cases, both sides of (2.6) are zero. The proof of Lemma 2.1 is then completed. We next give a well-known lemma which can be proved easily by scaling argument and simple computation. This simple result shows norm equivalence between the Sobolev seminorm and the discrete H 1 -seminorm.

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Lemma P 2.2. There exists a positive constant C0 > 0 independent of h such that M1 +M2 h vi φi ∈ SD (Ω) for any v = i=1 X C0−1 k∇vk20,2 ≤ |vi − vj |2 ≤ C0 k∇vk20,2 . (i,j)∈ω

Now we assume that A¯ is the L2 orthogonal projection of A onto (W h )4 ; i.e., Z 1 Aij (x)dx, 1 ≤ i, j ≤ 2, K ∈ Th . A¯ij |K = meas(K) K Moreover, define L1 (v) = −

M1 Z X

Z L2 (v) =

i=1

Ω

∂Vi

(A∇v) · nds,

(A∇v) · ∇vdx.

PM1 +M2 h vi φi ∈ SD (Ω) we have Then from Lemma 2.1, for any v = i=1 M Z Z 1 X ¯ ¯ ((A − A)∇v) · ndsvi − ((A − A)∇v) · ∇vdx |L1 (v) − L2 (v)| ≤ − ∂V Ω i i=1 Z M Z 1 X ¯ ¯ ((A − A)∇v) · ndsvi + ((A − A)∇v) · ∇vdx ≤ ∂Vi

i=1

Ω

= I1 + I2 and I2 ≤

X Z

X

K∈Th 1≤i,j≤2

However, |Aij (x) − A¯ij | ≤

1 meas(K)

Z

Consequently, I2 ≤Ch

K

K

|Aij − A¯ij ||∂i v||∂j v|dx.

|Aij (x) − Aij (x0 )|dx0 ≤ ChkAij k1,∞,Ω ,

X

X

x ∈ K.

kAij k1,∞,Ω k∇vk20,K

K∈Th 1≤i,j≤2

≤CkAk1,∞,Ω hk∇vk20,Ω . 0 denote the triangles with common edge zij . Then On the other hand, let Kij , Kij from Lemma 2.2 and the regularity condition, we know M1 Z X X ¯ ((A − A)∇v) · nij ds vi I1 = i=1 j∈ωi γij X Z ¯ ((A − A)∇v) · nij (vi − vj )ds = (i,j)∈ω,i