What Is a Good Linear Finite Element? Interpolation, Conditioning, Anisotropy, and Quality Measures (Preprint)

When a mesh of simplicial elements (triangles or tetrahedra) is used to form a piecewise linear approximation of a function, the accuracy of the approximation depends on the sizes and shapes of the elements. In finite element methods, the conditioning of the stiffness matrices also depends on the sizes and shapes of the elements. This article explains the mathematical connections between mesh geometry, interpolation errors, discretization errors, and stiffness matrix conditioning. These relationships are expressed by error bounds and element quality measures that determine the fitness of a triangle or tetrahedron for interpolation or for achieving low condition numbers. Unfortunately, the quality measures for these purposes do not fully agree with each other; for instance, small angles are bad for matrix conditioning but not for interpolation or discretization. The upper and lower bounds on interpolation error and element stiffness matrix conditioning given here are tighter than those usually seen in the literature, so the quality measures are likely to be unusually precise indicators of element fitness. Bounds are included for anisotropic cases, wherein long, thin elements perform better than equilateral ones. Surprisingly, there are circumstances wherein interpolation, conditioning, and discretization error are each best served by elements of different aspect ratios or orientations. Supported in part by the National Science Foundation under Awards ACI-9875170, CMS-9980063, CCR-0204377, and EIA9802069, and in part by a gift from the Okawa Foundation. The views and conclusions in this document are those of the author. They are not endorsed by, and do not necessarily reflect the position or policies of, the Okawa Foundation or the U. S. Government.