On Mesh Geometry and Stiffness Matrix Conditioning for General Finite Element Spaces

The performance of finite element computation depends strongly on the quality of the geometric mesh and the efficiency of the numerical solution of the linear systems resulting from the discretization of partial differential equation (PDE) models. It is common knowledge that mesh geometry affects not only the approximation error of the finite element solution but also the spectral properties of the corresponding stiffness matrix. In this paper, for typical second-order elliptic problems, some refined relationships between the spectral condition number of the stiffness matrix and the mesh geometry are established for general finite element spaces defined on simplicial meshes. The derivation of such relations for general high-order elements is based on a new trace formula for the element stiffness matrix. It is shown that a few universal geometric quantities have the same dominant effect on the stiffness matrix conditioning for different finite element spaces. These results provide guidance to the studies of both linear algebraic solvers and the unstructured geometric meshing.