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Analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem

Christoph Lehrenfeld, Tim van Beeck, Igor Voulis

Abstract

In this paper we present a new H(div)-conforming unfitted finite element method for the mixed Poisson problem which is robust in the cut configuration and preserves conservation properties of body-fitted finite element methods. The key is to formulate the divergence-constraint on the active mesh, instead of the physical domain, in order to obtain robustness with respect to cut configurations without the need for a stabilization that pollutes the mass balance. This change in the formulation results in a slight inconsistency, but does not affect the accuracy of the flux variable. By applying post-processings for the scalar variable, in virtue of classical local post-processings in body-fitted methods, we retain optimal convergence rates for both variables and even the superconvergence after post-processing of the scalar variable. We present the method and perform a rigorous a-priori error analysis of the method and discuss several variants and extensions. Numerical experiments confirm the theoretical results.

Analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem

Abstract

In this paper we present a new H(div)-conforming unfitted finite element method for the mixed Poisson problem which is robust in the cut configuration and preserves conservation properties of body-fitted finite element methods. The key is to formulate the divergence-constraint on the active mesh, instead of the physical domain, in order to obtain robustness with respect to cut configurations without the need for a stabilization that pollutes the mass balance. This change in the formulation results in a slight inconsistency, but does not affect the accuracy of the flux variable. By applying post-processings for the scalar variable, in virtue of classical local post-processings in body-fitted methods, we retain optimal convergence rates for both variables and even the superconvergence after post-processing of the scalar variable. We present the method and perform a rigorous a-priori error analysis of the method and discuss several variants and extensions. Numerical experiments confirm the theoretical results.
Paper Structure (44 sections, 15 theorems, 84 equations, 9 figures, 2 tables)

This paper contains 44 sections, 15 theorems, 84 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unfitted geometry, computational meshes and patches
  4. Notation for finite element spaces
  5. Notation for inner products, norms, inequalities up to constants
  6. Model problem
  7. Weak formulation
  8. Subproblems arising from a space decomposition
  9. Unfitted discretizations
  10. Restricted mixed FEM
  11. The discrete bilinear form $b_h(\cdot,\cdot)$
  12. The extended r.h.s. data $f_h$
  13. Ghost penalty stabilization
  14. The unfitted mixed FEM
  15. Consistency of the unfitted mixed FEM
  16. ...and 29 more sections

Key Result

Lemma 1

The method eq:new:mixedprob:a--eq:new:mixedprob:b with $f_h$ from eq:fh and $\gamma_f = \gamma_{\operatorname{div}}$ and the method eq:frachmethod from frachon2022divergence yield the same solution $u_h \in \Sigma_h$ and the same solution $p_h \in Q_h$ (and $\bar{p}_h \in Q_h$ respectively) away fro

Figures (9)

  • Figure 1: Example of an unfitted geometry, the active and cut part of the computational mesh (left) and a set of patches as in \ref{['ass:patches']}.
  • Figure 2: Background mesh (refinement level $0$) and cut domain for the considered geometry before (left) and after (right) parametric mesh deformation (for a mesh deformation of third order).
  • Figure 3: Numerical results for $\Vert u_h - u \Vert_{L^2}$ on $\Omega$ and $\Omega^{\mathcal{T}}$ with $\gamma_u = 0$ and $\gamma_u=1$ for polynomial degrees $k = 0,1,\dots,4$.
  • Figure 4: $L^2$-convergence of $\operatorname{div} u_h$ towards $-f$ (left) and $L^2$-error of $p_h - p$ on $\Omega$ (dashed) and $\Omega^{\text{int}}$ (solid) without post-processing (right) for polynomial degrees $k = 0,1,\dots,4$.
  • Figure 5: Post-processing results for $k=1$ (left) and $k=4$ (right).
  • ...and 4 more figures

Theorems & Definitions (40)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3: Patch-local integral conservation property
  • Remark 4
  • Lemma 2
  • proof
  • Remark 5
  • Remark 6
  • ...and 30 more