变度量各向异性网格生成算法和各向异性椭圆偏微分方程匹配网格

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

对于各向异性问题而言,在进行数值求解时,一个相匹配的各向异性网格至关重要.本文针对二阶椭圆偏微分方程,当其系数矩阵为各向异性时,给出了系数矩阵的逆作为相匹配的各向异性度量,在此度量下生成的各向异性网格即为匹配网格.本文还提出了新的变度量各向异性网格生成算法,该算法通过在各向异性背景网格上结合各向异性Delaunay准则和基于力平衡的结点优化函数生成高质量的各向异性三角化网格.通过数值算例可知,在匹配的各向异性三角化网格上,不仅有条件数小的代数离散系统,还有高精度的数值解,更甚者,在匹配网格结点上的l2范数误差有超收敛现象.

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 l2 norm on the mesh nodes. Various numerical examples are presented to demonstrate the effectiveness of the proposed method and natural metric.