What is a Good Linear Element? Interpolation, Conditioning, and Quality Measures

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 paper explains the mathematical connections between mesh geometry, interpolation 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 two purposes do not agree with each other; for instance, small angles are bad for matrix conditioning but not for interpolation. Several of the upper and lower bounds on interpolation errors and element stiffness matrix conditioning given here are tighter than those that have appeared in the literature before, so the quality measures are likely to be unusually precise indicators of element fitness.