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

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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.

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, Guangdong, China

    Haitao Leng

  2. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Xiamen, 361005, Fujian, China

    Huangxin Chen

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  1. Haitao Leng
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Correspondence to Huangxin Chen.

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The work of Haitao Leng was supported by the Cultivation Project of SCNU (Grant No. 19KJ08) and the NSF of China (Grant No. 12001209). The work of Huangxin Chen was supported by the NSF of China (Grant No. 11771363) and the Fundamental Research Funds for the Central Universities (Grant No. 20720180003).

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Leng, H., Chen, H. Adaptive HDG Methods for the Brinkman Equations with Application to Optimal Control. J Sci Comput 87, 46 (2021). https://doi.org/10.1007/s10915-021-01450-x

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