The curved Mimetic Finite Difference method: allowing grids with curved faces

TL;DR

This paper introduces a novel mimetic finite difference method that effectively handles diffusion problems on grids with curved faces, ensuring symmetry and optimal convergence with minimal unknowns per face.

Contribution

It proposes a new approach based on $P_{0}$-consistency, allowing for accurate and symmetric discretization on non-planar face grids, which was not possible with standard methods.

Findings

Method converges with optimal rate on curved and perturbed grids.

Uses only one unknown per curved face, simplifying computations.

Numerical tests confirm theoretical properties.

Abstract

We present a new mimetic finite difference method for diffusion problems that converges on grids with \textit{curved} (i.e., non-planar) faces. Crucially, it gives a symmetric discrete problem that uses only one discrete unknown per curved face. The principle at the core of our construction is to abandon the standard definition of local consistency of mimetic finite difference methods. Instead, we exploit the novel and global concept of $P_{0}$-consistency. Numerical examples confirm the consistency and the optimal convergence rate of the proposed mimetic method for cubic grids with randomly perturbed nodes as well as grids with curved boundaries.