Exact a posteriori error analysis of the least squares finite element method

Abstract

A residual type a posteriori error estimator is presented for the least squares finite element method. The estimator is proved to equal the exact error in a norm induced by some least squares functional. The error indicator of each element is equal to the exact error norm restricted to the element as well. In other words, the estimator is perfectly effective and reliable for error control and for adaptive mesh refinement. The exactness property requires virtually no assumptions on the regularity of the solution and on the finite element order in the approximation or in the estimation. The least squares method is in a very general setting that applies to various linear boundary-value problems such as the elliptic systems of first-order and of even-order and the mixed type partial differential equations. Numerical results are given to demonstrate the exactness. (C) 2000 Elsevier Science Inc. All rights reserved.