A monotone $Q^1$ finite element method for anisotropic elliptic equations

TL;DR

This paper introduces a new monotone finite element method for anisotropic elliptic equations that ensures the discrete maximum principle and is proven to converge, with specific mesh constraints for quadrilateral meshes.

Contribution

It develops a monotone continuous $Q^1$ finite element scheme for anisotropic diffusion with proven convergence and mesh constraints for quadrilateral meshes.

Findings

The scheme guarantees the discrete maximum principle.

Convergence of the method is rigorously proven.

Mesh constraints are identified for quadrilateral meshes.

Abstract

We construct a monotone continuous $Q^1$ finite element method on the uniform mesh for the anisotropic diffusion problem with a diagonally dominant diffusion coefficient matrix. The monotonicity implies the discrete maximum principle. Convergence of the new scheme is rigorously proven. On quadrilateral meshes, the matrix coefficient conditions translate into specific mesh constraints.