Adjoint-based anisotropic hp-adaptation for discontinuous Galerkin methods using a continuous mesh model

In this paper we propose an adjoint-based hp-adaptation method for conservation laws, and corresponding numerical schemes based on piecewise polynomial approximation spaces. The method uses a continuous mesh framework, similar to that proposed in [1], where a global optimization scheme was formulated with respect to the error in the numerical solution, measured in any L-q norm. The novelty of the present work is the extension to more general optimization targets. Here, any solution-dependent functional, which is compatible with an adjoint equation, may be the target of the continuous-mesh optimization. We present the rationale behind the formulation of the optimization problem, with particular emphasis on the continuous mesh model, and the relevant adjoint-based error estimate. Additionally we combine the adjoint-based error estimates with the polynomial optimization strategy from [2] to present a complete hp-adaptation method which shows exponential convergence in the target function. The h-only mesh adaptation strategy of this work has been presented as a conference proceeding earlier [3]. Numerical experiments are carried out to demonstrate the viability of the scheme. (C) 2020 Elsevier Inc. All rights reserved.