Finite Element Approximation for First Order Systems

Abstract

In this paper we examine a finite element procedure for the numerical approximation of the solution of certain classes of boundary value problems for first order partial differential equations. Our first result shows, under.a weak regularity assumption, that the error in the $L_2 $ norm is $O(h^{K - 1} )$ for piecewise polynomials of degree $K - 1$; thus, convergence is not optimal in $L_2 $. We show by an example that there is, in fact, a loss in $L_2 $ norm. However, under stronger regularity assumptions, which apply primarily to elliptic systems, we show that convergence in $L_2 $ is optimal, e.g., with piecewise linear elements the error is of $O(h^2 )$. Numerical examples indicating the effectiveness of the method are given.