[Paper Review] Discrete maximum principle and a Delaunay-type mesh condition for linear finite element approximations of two-dimensional anisotropic diffusion problems

Weizhang Huang|arXiv (Cornell University)|Aug 3, 2010

Advanced Numerical Methods in Computational Mathematics4 citations

TL;DR

This paper introduces a Delaunay-type mesh condition for linear finite element methods solving two-dimensional anisotropic diffusion problems, ensuring satisfaction of the discrete maximum principle (DMP). The condition generalizes the classical Delaunay criterion to anisotropic diffusion by using a metric derived from the inverse diffusion matrix, and it is proven to be weaker than the existing anisotropic non-obtuse angle condition, enabling monotone solutions on coarser, more flexible meshes.

ABSTRACT

The finite element solution of two-dimensional anisotropic diffusion problems is considered. A Delaunay-type mesh condition is developed for linear finite element approximations to satisfy a discrete maximum principle. The condition is shown to be weaker than the existing anisotropic non-obtuse angle condition. It reduces to the well known Delaunay condition for the special case with the identity diffusion matrix. Numerical results are presented to verify the theoretical findings.

Motivation & Objective

• To develop a mesh condition that ensures linear finite element approximations satisfy the discrete maximum principle (DMP) for two-dimensional anisotropic diffusion problems.
• To generalize the classical Delaunay condition to anisotropic diffusion by incorporating the diffusion matrix into the geometric criterion.
• To establish a mesh condition that is weaker than the anisotropic non-obtuse angle condition, allowing for more flexible and coarser meshes while preserving monotonicity.
• To provide a theoretical foundation based on the global stiffness matrix, rather than local element analysis, to derive the condition.
• To verify the theoretical findings through numerical experiments on meshes with varying geometric and anisotropic properties.

Proposed method

• Derives a new mesh condition by analyzing the global stiffness matrix of the linear finite element formulation, rather than relying on local element stiffness matrices.
• Introduces a metric-dependent angle condition using the inverse diffusion matrix $\mathbb{D}^{-1}$, defining angles between edges in the $\mathbb{D}^{-1}$-norm.
• Proposes a Delaunay-type condition requiring the sum of angles opposite a common edge in the $\mathbb{D}^{-1}$-norm to be at most $\pi$ for all element pairs.
• Establishes the condition as a sufficient condition for DMP satisfaction in linear finite element approximations of anisotropic diffusion problems.
• Applies numerical quadrature with consistent weights and nodal points to approximate the weak form, ensuring consistency with the theoretical formulation.
• Uses edge-swapping procedures to generate meshes satisfying the new condition, leveraging known Delaunay mesh generation techniques in the $\mathbb{D}^{-1}$-metric.

Experimental results

Research questions

• RQ1 Can a Delaunay-type mesh condition be formulated for linear finite element methods to ensure discrete maximum principle satisfaction in anisotropic diffusion problems?
• RQ2 How does the new mesh condition compare in strength to the existing anisotropic non-obtuse angle condition in terms of mesh flexibility and monotonicity preservation?
• RQ3 Does the proposed condition reduce to the classical Delaunay condition when the diffusion matrix is the identity?
• RQ4 Can the new condition be used to guide the design of edge-swapping algorithms for monotone finite element solutions in anisotropic diffusion?
• RQ5 What is the convergence behavior of undershoots and overshoots in numerical solutions when the mesh does not satisfy the new condition?

Key findings

• The proposed Delaunay-type mesh condition is strictly weaker than the anisotropic non-obtuse angle condition, allowing for a broader class of meshes to satisfy DMP.
• For constant diffusion matrices, the condition reduces to $\alpha_{ij,\mathbb{D}^{-1}}^{K} + \alpha_{ij,\mathbb{D}^{-1}}^{K'} \leq \pi$, which generalizes the classical Delaunay condition.
• Numerical results show that meshes satisfying the new condition (e.g., Fig. 4b, 4c) produce finite element solutions that remain between 0 and 1, with no undershoots or overshoots.
• Meshes violating the condition (e.g., Fig. 4a, 4d) produce solutions with significant undershoots and overshoots, even as the mesh is refined.
• The magnitude of undershoots and overshoots decays slowly at a rate of $O(N^{-0.5})$ as the number of elements $N$ increases, confirming the lack of monotonicity when the condition is violated.
• The condition is geometrically interpretable and can be used as a criterion for edge-swapping algorithms to generate monotone, anisotropy-aware meshes.