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The Tempered Finite Element Method

Antoine Quiriny, Václav Kučera, Jonathan Lambrechts, Nicolas Moës, Jean-François Remacle

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.

The Tempered Finite Element Method

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.

Paper Structure

This paper contains 18 sections, 14 theorems, 106 equations, 22 figures.

Table of Contents

  1. Introduction
  2. A simple numerical experiment
  3. Finite element formulation of the 2D Poisson problem
  4. Tempered finite element on a degenerate mesh
  5. Convergence
  6. Error sensitivity and choice of
  7. Size effect related to the degenerate band/surface
  8. Theory behind the TFEM scheme
  9. TFEM for a band of arbitrarily small but non-zero caps
  10. Zero-measure elements -- TFEM and Mortaring
  11. Extensions
  12. Extension to linear elasticity
  13. Mesh mortaring
  14. High order elements
  15. Advection
  16. ...and 3 more sections

Key Result

Lemma 1

Let $K\subset\mathbb{R}^2$ be an arbitrary triangle. Let $u\in W^{2,\infty}(K)$ and let $\Pi_K u$ be the linear Lagrange interpolation of $u$ on $K$. Then there exists a constant $C_c$ independent of $u$ and $K$ such that where $R_K$ is the circumradius of the $K$ triangle.

Figures (22)

  • Figure 1: A triangle $T$. The angle $\theta_i$ is associated to the vertex $x_i$.
  • Figure 2: Band of caps triangles in 2D.
  • Figure 3: Mapping between the physical and reference domains.
  • Figure 4: Three 2D meshes: regular (left), with a band of caps (center) and with a fully degenerated band of caps (right).
  • Figure 5: Solution of the manufactured solution on the regular mesh and on a mesh with a degenerated band with $h=\frac{1}{10}$ for different values of $J_{\text{min}}$.
  • ...and 17 more figures

Theorems & Definitions (27)

  • Lemma 1: Circumradius estimate kobayashi2015circumradius
  • Lemma 2: Maximum-angle condition babuvska1976angle
  • Lemma 3: Young's inequality
  • Theorem 4
  • proof
  • Remark 1
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • ...and 17 more