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A monotone $Q^1$ finite element method for anisotropic elliptic equations

Hao Li, Xiangxiong Zhang

TL;DR

The paper addresses solving heterogeneous anisotropic diffusion equations with a diffusion matrix $\mathbf{a}(\mathbf{x})$ while preserving a discrete maximum principle. It proposes a linear, second-order monotone $Q^1$ finite element method on a uniform mesh, using a mixed quadrature to induce an $M$-matrix structure and prove convergence via $V^h$-ellipticity and duality arguments. A key contribution is the derivation of explicit per-element coefficient and mesh constraints that guarantee monotonicity, plus an extension to general quadrilateral meshes under local mesh conditions, validated by comprehensive numerical tests on both uniform and quad-based meshes. The method provides a robust, oscillation-free approach for anisotropic diffusion with rigorous error bounds and practical guidance for mesh design in heterogeneous media.

Abstract

We construct a monotone continuous $Q^1$ finite element method on the uniform mesh for the anisotropic diffusion problem with a diagonally dominant diffusion coefficient matrix. The monotonicity implies the discrete maximum principle. Convergence of the new scheme is rigorously proven. On quadrilateral meshes, the matrix coefficient conditions translate into specific mesh constraints.

A monotone $Q^1$ finite element method for anisotropic elliptic equations

TL;DR

The paper addresses solving heterogeneous anisotropic diffusion equations with a diffusion matrix while preserving a discrete maximum principle. It proposes a linear, second-order monotone finite element method on a uniform mesh, using a mixed quadrature to induce an -matrix structure and prove convergence via -ellipticity and duality arguments. A key contribution is the derivation of explicit per-element coefficient and mesh constraints that guarantee monotonicity, plus an extension to general quadrilateral meshes under local mesh conditions, validated by comprehensive numerical tests on both uniform and quad-based meshes. The method provides a robust, oscillation-free approach for anisotropic diffusion with rigorous error bounds and practical guidance for mesh design in heterogeneous media.

Abstract

We construct a monotone continuous finite element method on the uniform mesh for the anisotropic diffusion problem with a diagonally dominant diffusion coefficient matrix. The monotonicity implies the discrete maximum principle. Convergence of the new scheme is rigorously proven. On quadrilateral meshes, the matrix coefficient conditions translate into specific mesh constraints.
Paper Structure (21 sections, 15 theorems, 112 equations, 2 figures, 6 tables)

This paper contains 21 sections, 15 theorems, 112 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Monotonicity and discrete maximum principle
  3. Monotone schemes for anisotropic diffusion equations
  4. Contributions and organization of the paper
  5. Preliminaries
  6. Notation and tools
  7. Mixed quadrature
  8. Quadrature error estimates
  9. The $Q^1$ finite element method and its monotonicity
  10. Derivation of the scheme
  11. Discrete maximum principle
  12. Monotonicity of the $Q^1$ finite element
  13. Convergence of the $Q^1$ finite element method with mixed quadrature
  14. Approximation error estimate of bilinear forms
  15. $V^h$-ellipticity and the dual problem
  16. ...and 6 more sections

Key Result

Theorem 1

\newlabelbh-lemma0 If a continuous linear mapping $\hat{\Pi}: H^{k+1}(\hat{K})\rightarrow H^{k+1}(\hat{K})$ satisfies $\hat{\Pi} \hat{v}=\hat{v}$ for any $\hat{v}\in Q^k(\hat{K})$, then Therefore if $l(\cdot)$ is a continuous linear form on the space $H^{k+1}(\hat{K})$ satisfying $l(\hat{v})=0, \,\forall \hat{v}\in Q^k(\hat{K}),$ then where $\|l\|'_{k+1, \hat{K}}$ is the norm in the dual space

Figures (2)

  • Figure 1: A quadrilateral element $e$.
  • Figure 1: Quadrilateral mesh.

Theorems & Definitions (34)

  • Theorem 1
  • Lemma 2
  • Proof 1
  • Lemma 3
  • Proof 2
  • Lemma 4
  • Proof 3
  • Lemma 5
  • Proof 4
  • Theorem 1
  • ...and 24 more