Error estimates for anisotropic finite elements and applications

Abstract

The finite element method is one of the most frequently used techniques to approximate the solution of partial differential equations. It consists in approximating the unknown solution by functions which are polynomials on each element of a given partition of the domain, made of triangles or quadrilaterals (or their generalizations to higher dimensions).

A fundamental problem is to estimate the error between the exact solution u and its computable finite element approximation. In many situations this error can be bounded in terms of the best approximation of u by functions in the finite element space of piecewise polynomial functions. A natural way to estimate this best approximation is by means of the Lagrange interpolation or other similar procedures.

Many works have considered the problem of interpolation error estimates. The classical error analysis for interpolations is based on the so-called regularity assumption, which excludes elements with different sizes in each direction (called anisotropic). The goal of this paper is to present a different approach which has been developed by many authors and can be applied to obtain error estimates for several interpolations under more general hypotheses.

An important case in which anisotropic elements arise naturally is in the approximation of convection-diffusion problems which present boundary layers. We present some applications to these problems.

Finally we consider the finite element approximation of the Stokes equations and present some results for non-conforming methods.