Goal-oriented spatial adaptivity for the SN equations on unstructured triangular meshes

Abstract

In this paper we investigate a goal-oriented adaptive algorithm for the S N equations on unstructured meshes. The method is based on a dual-weighted residual approach where an appropriate adjoint problem is formulated and solved in order to obtain the importance of residual errors in the forward problem on the specific goal of interest. The forward residuals and the adjoint function are combined to obtain both economical finite element meshes tailored to the solution of the target functional as well as providing reliable error estimates. The suitability of the approach is demonstrated by performing a series of numerical experiments in two shielding geometries exhibiting strong heterogeneity and flux gradients. Tests are performed with varying target functionals. The effectiveness of the method is shown by comparing the number of elements required for a specified accuracy with that necessary using uniform refinement to give an indication what savings can be achieved by adaptive refinement.