Optimally Adapted Meshes for Finite Elements of Arbitrary Order and W1p Norms

Abstract

Given a function f defined on a bidimensional bounded domain and a positive integer N, we study the properties of the triangulation that minimizes the distance between f and its interpolation on the associated finite element space, over all triangulations of at most N elements. The error is studied in the W1p norm and we consider Lagrange finite elements of arbitrary polynomial order m-1. We establish sharp asymptotic error estimates as N tends to infinity when the optimal anisotropic triangulation is used. A similar problem has been studied earlier, but with the error measured in the Lp norm. The extension of this analysis to the W1p norm is crucial in order to match more closely the needs of numerical PDE analysis, and it is not straightforward. In particular, the meshes which satisfy the optimal error estimate are characterized by a metric describing the local aspect ratio of each triangle and by a geometric constraint on their maximal angle, a second feature that does not appear for the Lp error norm. Our analysis also provides with practical strategies for designing meshes such that the interpolation error satisfies the optimal estimate up to a fixed multiplicative constant. We discuss the extension of our results to finite elements on simplicial partitions of a domain of arbitrary dimension, and we provide with some numerical illustration in two dimensions.