The Tempered Finite Element Method

TL;DR

The paper introduces the Tempered Finite Element Method (TFEM), extending FEM to handle meshes with degenerate elements, ensuring convergence where standard FEM fails, with theoretical proofs and practical extensions.

Contribution

It proposes a simple, implementable modification to FEM that guarantees convergence on problematic meshes, broadening FEM applicability.

Findings

Proves convergence of TFEM for zero-measure elements

Demonstrates TFEM's effectiveness in linear elasticity and non-conforming meshes

Shows TFEM's compatibility with high-order elements and advection

Abstract

In this paper, we propose a new approach -- the Tempered Finite Element Method (TFEM) -- that extends the Finite Element Method (FEM) to classes of meshes that include zero-measure or nearly degenerate elements for which standard FEM approaches do not allow convergence. First, we review why the maximum angle condition [2] is not necessary for FEM convergence and what are the real limitations in terms of meshes. Next, we propose a simple modification of the classical FEM for elliptic problems that provably allows convergence for a wider class of meshes including bands of caps that cause locking of the solution in standard FEM formulations. The proposed method is trivial to implement in an existing FEM code and can be theoretically analyzed. We prove that in the case of exactly zero-measure elements it corresponds to mortaring. We show numerically and theoretically that what we propose is functional and sound. The remainder of the paper is devoted to extensions of the TFEM method to linear elasticity, mortaring of non-conforming meshes, high-order elements, and advection.