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On the accuracy of the finite volume element method for diffusion equations on composite grids. (English) Zbl 0707.65073

Consider the elliptic problem -(A(x,y)w)=f(x,y) in Ω=(0,1) 2 , w=0 on Ω, which is equivalent with: Find wH 0 2 (Ω) such that, for any admissible volume VΩ ¯

(1)- V (A(x,y)x)n dS= V fdV·

Then, the finite volume element (FVE) method for approximating the solution (1) consists of defining a similar problem in a finite-dimensional subspace UH 0 1 (Ω) for a finite set of volumes {V αβ } (α,β) , (α,β)S, for a given S: Find uU such that

(2)(α,β)S,- V αβ (A(x,y)u)n dS= V αβ fdV·

We denote e(p)=u(p)-w(p), the discretization error; p=(x α ,y β )G, where u and w are the solutions of (1) and (2), respectively, and G is the composite grid, GΩ. A first evaluation error result is obtained by: If wH 0 m (Ω) and AW m-1 C m-2 (Ω), m=2 or 3, then:

e 1,G C((2h) m-1 w m,Ω-Ω F +h m/2 w m,Ω F )·

An improved error is given by: If wH 0 m (Ω), where m=2 or 3, then:

e 1,G C(2h) m-1 |w| m,ΩΩ F + +h m-1 |w| m,Ω F ),

where C is a constant independent of the mesh size h,· 1,G ; · m, ; |·| m, ; are the Sobolev norm and seminorm, respectively, implicitly given in the paper. The paper contains a detailed presentation of the FVE method.

Reviewer: T.Potra
MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65N38Boundary element methods (BVP of PDE)
65N15Error bounds (BVP of PDE)
35J25Second order elliptic equations, boundary value problems