Total flux estimates for a finite element approximation of the Dirichlet problem using the boundary penalty method

Abstract

This paper considers a fully practical piecewise linear finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation,Au=f, in a smooth regionΩ?<? n (n=2 or 3) by the boundary penalty method. Using an unfitted mesh; that isΩ h , an approximation of Ω with dist (Ω,Ω h )?Ch 2 is not in general a union of elements; and assumingu?H 4 (Ω) we show that one can recover the total flux across a segment of the boundary of Ω with an error ofO(h 2). We use these results to study a fully practical piecewise linear finite element approximation of an elliptic equation by the boundary penalty method when the prescribed data on part of the boundary is the total flux.