Recovered finite element methods on polygonal and polyhedral meshes

TL;DR

This paper extends recovered finite element methods (R-FEM) to polygonal and polyhedral meshes, providing conforming discretizations with efficiency comparable to discontinuous Galerkin methods, supported by theoretical error bounds and numerical validation.

Contribution

The paper introduces a novel extension of R-FEM to general polygonal and polyhedral meshes, enabling conforming discretizations with fewer degrees of freedom than traditional methods.

Findings

A priori error bounds established for general linear second order problems.

Numerical experiments demonstrate good practical performance.

Framework effectively handles complex mesh geometries.

Abstract

Recovered finite element methods (R-FEM) have been recently introduced for meshes consisting of simplicial and/or box-type meshes. Here, utilising the flexibility of R-FEM framework, we extend their definition on polygonal and polyhedral meshes in two and three spatial dimensions, respectively. A key attractive feature of this framework is its ability to produce conforming discretizations, yet involving only as many degrees of freedom as discontinuous Galerkin methods over general polygonal/polyhedral meshes with potentially many faces per element. A priori error bounds are shown for general linear, possibly degenerate, second order advection-diffusion-reaction boundary value problems. A series of numerical experiments highlights the good practical performance of the proposed numerical framework.