An adjoint-based adaptive error approximation of functionals by the hybridizable discontinuous Galerkin method for second-order elliptic equations

Abstract

This paper presents a novel, fully computable error approximation and mesh adaptation approach for functionals defined by second-order elliptic equations. The functionals are approximated by the hybridizable discontinuous Galerkin (HDG) method and the error approximation is obtained by the adjoint-based method and a local post-processing technique of the HDG method. Unlike most adjoint-based error estimations in the literature, the novelty of our method is that the error approximation is obtained without requiring an auxiliary finer mesh or higher order approximation spaces for solving the adjoint problem. This reduces the computational cost and eases the implementation of the problem. What's more, the local post-processing technique can be carried out in parallel, which speeds up the method even more. We illustrate the method with a second-order elliptic problem and we present examples of three types of functionals: volume integrals; boundary integrals and eigenvalue problems. Numerical tests with adaptive mesh refinements for non-smooth solutions are presented to show that our method is efficient and robust.