Output‐based mesh optimization for hybridized and embedded discontinuous Galerkin methods

Abstract

SummaryThis paper presents a method for optimizing computational meshes for the prediction of scalar outputs when using hybridized and embedded discontinuous Galerkin (HDG/EDG) discretizations. Hybridization offers memory and computational time advantages compared to the standard discontinuous Galerkin (DG) method through a decoupling of elemental degrees of freedom and the introduction of face degrees of freedom that become the only globally coupled unknowns. However, the additional equations of weak flux continuity on each interior face introduce new residuals that augment output error estimates and complicate existing element‐centric mesh optimization methods. This work presents techniques for converting face‐based error estimates to elements and sampling their reduction with refinement in order to determine element‐specific anisotropic convergence rate tensors. The error sampling uses fine‐space adjoint projections and does not require additional solves on subelements. Together with a degree‐of‐freedom cost model, the error models drive metric‐based unstructured mesh optimization. Adaptive results for inviscid and viscous two‐dimensional flow problems demonstrate (i) improvement of EDG mesh optimality when using error models that incorporate face errors, (ii) the relative insensitivity of HDG mesh optimality to the incorporation of face errors, and (iii) degree of freedom and computational‐time benefits of hybridized methods, particularly EDG, relative to DG.