Asymptotic a posteriori finite element bounds for the outputs of noncoercive problems: the Helmholtz and Burgers equations

We describe an a posteriori finite element procedure for the efficient comp utation of lower and upper estimators for linear-functional outputs of nonc oercive linear and semilinear elliptic second-order partial differential eq uations. Under a relatively weak hypothesis related to the relative magnitu de of the L-2 and H-1 errors of the reconstructed solution, these lower and upper estimators converge to the true output from below and above, respect ively, and thus constitute asymptotic bounds. In numerical experiments we f ind that our hypothesis is satisfied once the finite element triangulation even roughly resolves the structure of the exact solution, and thus, in pra ctice, the bounds prove quite reliable. Numerical results are presented for the one-dimensional Helmholtz equation and for the Burgers equation. (C) 1 999 Elsevier Science S.A. All rights reserved.