Quantitative Error Control for Finite Element Methods for One-Dimensional Compressible Flow

Abstract

We prove a posteriori $L_{2}(L_{2})$ and $L_{\infty}(H^{-1})$ residual based error estimates. The estimates contain certain strong stability factors or weights related to the solution of a linearized dual problem associated with the Euler equations. We compute the stability factors and weights by solving the dual problem numerically and show that quantitative error control is possible for one-dimensional (1-D) compressible flow.