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Finite volume approximation of elliptic problems and convergence of an approximate gradient. (English) Zbl 0982.65122
The authors extend earlier work [for example in Numer. Math. 82, No. 1, 91-116 (1999; Zbl 0930.65118)] on the convergence of finite volume solution methods for second order partial differential equations. The current work adds convergence of the solution gradient to earlier proofs of convergence of the solution alone.
The particular finite volume method under study requires an “admissible” mesh that depends on the line connecting the centers of adjacent mesh cells being perpendicular to the common edge of the cells. Triangular and Voronoi meshes are examples of admissible meshes. Degrees of freedom of the solution are then defined not only for mesh cells, but also for mesh edges. The mesh edge degrees of freedom are then used to define the gradient of the solution. For a mesh cell $$K$$ with one edge $$\sigma$$, the solution is regarded as constant over $$K$$, but its gradient is regarded as being defined in terms of functions $$\phi$$ that satisfy a Neumann problem with normal derivative equal to 1 on $$\sigma$$ and zero on the other edges of $$K$$. The properties of $$\phi$$ play a crucial role in the convergence estimates.
Two numerical examples are presented for the Dirichlet problem for Poisson’s equation. The first involves a spatially variable density and is posed on a square two-dimensional domain, and the second involves Poisson’s equation on an “$$L$$”-shaped domain.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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