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

Abstract

We describe an a posteriori finite element procedure for the efficient computation of lower and upper estimators for linear-functional outputs of noncoercive linear and semilinear elliptic second-order partial differential equations. Under a relatively weak hypothesis related to the relative magnitude 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, respectively, and thus constitute asymptotic bounds. In numerical experiments we find that our hypothesis is satisfied once the finite element triangulation even roughly resolves the structure of the exact solution, and thus, in practice, the bounds prove quite reliable. Numerical results are presented for the one-dimensional Helmholtz equation and for the Burgers equation.