Adjoint A Posteriori Error Measures for Anisotropic Mesh Optimisation

Abstract

In this paper an adjoint- (or sensitivity-) based error measure is formulated which measures the error contribution of each solution variable to an overall goal The goal is typically embodied in an integral functional, e.g., the solution in a small region of the domain of interest. The resulting a posteriori error measures involve the solution of both primal and adjoint problems. A comparison of a number of important a posteriori error measures is made in this work. There is a focus on developing relatively simple methods that refer to information from the discretised equation sets (often readily accessible in simulation codes) and do not explicitly use equation residuals. This method is subsequently used to guide anisotropic mesh adaptivity of tetrahedral finite elements. Mesh adaptivity is achieved here with a series of optimisation heuristics of the landscape defined by mesh quality. Mesh quality is gauged with respect to a Riemann metric tensor embodying an a posteriori error measure, such that an ideal element has sides of unit length when measured with respect to this metric tensor. This results in meshes in which each finite-element node has approximately equal (subject to certain boundary-conforming constraints and the performance of the mesh optimisation heuristics) error contribution to the functional (goal).