Error sensitivity to refinement: a criterion for optimal grid adaptation

Abstract

Most indicators used for automatic grid refinement are suboptimal, in the sense that they do not really minimize the global solution error. This paper concerns with a new indicator, related to the sensitivity map of global stability problems, suitable for an optimal grid refinement that minimizes the global solution error. The new criterion is derived from the properties of the adjoint operator and provides a map of the sensitivity of the global error (or its estimate) to a local mesh refinement. Examples are presented for both a scalar partial differential equation and for the system of Navier–Stokes equations. In the last case, we also present a grid-adaptation algorithm based on the new estimator and on the FreeFem+ + software that improves the accuracy of the solution of almost two order of magnitude by redistributing the nodes of the initial computational mesh.