Adaptive HDG Methods for the Brinkman Equations with Application to Optimal Control

Abstract

This paper investigates adaptive hybridizable discontinuous Galerkin methods for the gradient-velocity–pressure formulation of Brinkman equations. We use piecewise polynomials of degree k to approximate the velocity, the velocity gradient, the pressure and the boundary traces. By the $$L^2$$ -projection and inf-sup condition, a residual type a posteriori error estimator is introduced for the error measured by an energy norm. Moreover, we prove that the a posteriori error estimator is robust in the sense that the ratio of the upper and lower bounds is independent of the dynamic viscosity and permeability. Then the idea and the result, which have been used and obtained for Brinkman equations, are extended to solve the Brinkman optimal control problem. Based on the variational discretization concept, we also derive an efficient and reliable a posteriori error estimator for the error measured by an energy norm. Finally, several numerical results are provided to validate the theoretical analysis.