A monotone finite volume element scheme for diffusion equations on triangular grids

Abstract

We present a monotone finite volume element scheme for the diffusion problem on triangular grids, which stems from some idea about the two-point flux in the literature [15]. A new nonlinear two-point flux formulation is obtained for some part of one control volume edge, and this straight edge crosses through some common edge of two neighboring triangular elements where two inner points from these neighboring elements are chosen and linked so that it does not cross through other triangle elements even if the meshes are severely distorted. The approximation of κ∇u in some quadrilateral region including this straight control volume edge, are obtained by some weighted average with the gradient of some finite element functions, and the weighted coefficients are determined by the nonlinear two-point flux idea. Furthermore, we prove that the new scheme is monotone under some conditions that one modified barycenter dual partition Ωh⁎ is chosen for some given partition Ωh and the exact solution u∈C1(Ω), and the corresponding coefficient matrix is of M-matrix. Finally, we carry on some typical experiments. Numerical results verify the monotonicity of this new scheme, and also confirm the accuracy and stability.