RECENT EXPERIENCES WITH ERROR ESTIMATION AND ADAPTIVITY .1. REVIEW OF ERROR ESTIMATORS FOR SCALAR ELLIPTIC PROBLEMS

In this paper we review two classes of methods for the a posteriori estimation of the error in finite element approximations of elliptic boundary value problems, namely residual and flux-projection error estimators. Several versions of the residual method and a popular version of the flux projection method were implemented for approximations of a scalar elliptic model problem defined on uniform grids of triangles with hierarchic shape functions of polynomial order p (1 less-than-or-equal-to p less-than-or-equal-to 7). Numerical experiments indicate that flux projection schemes, which are now in use in several commercial finite element codes, are not asymptotically exact and may lead to very poor predictions of the error for even-order finite element approximations. Fortunately residual estimates, which are theoretically justified, provide asymptotically exact estimators of the global energy norm of the error. Both classes of error estimates may fail to give uniformly accurate estimates of the local distribution of error over the grid.