Anisotropic mesh generation methods with variable metrics and natural metric for anisotropic elliptic equation

Abstract

A suitable anisotropic mesh is important for the anisotropic problem in many applications. Given a suitable metric tensor and using an efficient algorithm to generate the anisotropic mesh are crucial steps in the anisotropic mesh methodology. In this paper, we develop a new anisotropic mesh generation method by combining the modified anisotropic Delaunay criterion and force-based equilibrium smoothing function on an anisotropic background mesh. We observe the natural metric for the anisotropic problem with a variable anisotropic diffusion is the inverse of the diffusion in three beneficial aspects: Better discrete algebraic systems with smaller condition numbers, more accurate finite element solution and superconvergence in $l^2$ norm on the mesh nodes. Various numerical examples are presented to demonstrate the effectiveness of the proposed method and natural metric.