A discontinuous Galerkin goal-oriented error estimator based on orthogonal discrete dual spaces

Goal-oriented error estimation [1] has seen increasing interest for adaptive discretizations of engineering problems, as these methods aim to minimize the error in the quantity of interest that motivates the engineering analysis in the first place. However, the main disadvantage of deriving refinement indicators based on goal-oriented error estimators is that an additional “dual problem” needs to be solved, often in higher resolution than the primal problem, comparing badly with the simpler estimators requiring only local operations in terms of the computational cost.In this work, we aim to reduce the cost of approximating the dual solution. Taking inspiration from the fact that often its interpolation on the primal mesh is subtracted from the approximate dual solution in obtaining localized refinement indicators, we propose to discretize the dual problem using a space orthogonal to the primal discretization space. We assume to start with a discrete dual space that is a strict superset of the primal discretization space and decompose it into coarse and fine parts and only use the fine-scale solution, discarding the coarse problem. We then apply the ideas originating from the variational multiscale framework [2] to take care of the influence of the discarded coarse scales on the finer resolved scales. We prove that under certain conditions the fine-coarse coupling can be chosen to disappear altogether, resulting in a cheap error estimator identical to the conventional and more expensive estimator based on dual discretizations using a refinement of the discrete primal spaces.We choose to apply these concepts to the discontinuous Galerkin methods [3], leveraging the simplicity of constructing orthogonal bases for discontinuous polynomial spaces. We numerically investigate the quality of the resulting estimator for a 1D convection-diffusion problem with varying Peclet numbers. Our preliminary results indicate that the proposed estimator performs exceptionally in the diffusion-dominated range; however, the dependence on the choice of the projection in splitting the scales and whether or not a coarse scale model is employed appears to be very significant in the advection-dominated regime.