## Convergence of mimetic finite difference discretizations of the diffusion equation.(English)Zbl 1014.65114

The Dirichlet problem on a bounded polygonal convex domain in $$R^2$$ is considered for the linear diffusion equation. The authors rewrite the equation in the standard way as a system of two first order equations for which the theory of mixed finite element methods is usually applied. They show that some special grid methods on triangular and quadrilateral grids can be treated as perturbations of mixed finite element methods introduced by P. A. Raviart and J. M. Thomas [Lect. Notes Math. 606, 292-315 (1977; Zbl 0362.65089)].
The main mathematical result is connected with the proof that the methods under consideration have practically the same accuracy as the original mixed finite element methods. Numerical examples are presented as well for the unit square and grids of the simplest topological structure with $$h\in [2^{-9},2^{-4}]$$.

### MSC:

 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N06 Finite difference methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Zbl 0362.65089