Abstract
SummaryIn this paper, we present an adaptive mesh refinement method for solving convection‐diffusion‐reaction equations on surfaces, which is a fundamental subproblem in many models for simulating the transport of substances on biological films and solid surfaces. The method considered is a combination of well‐known techniques: the surface finite element method, streamline diffusion stabilization, and the gradient recovery–based Zienkiewicz‐Zhu error estimator. The streamline diffusion method overcomes the instability issue of the finite element method for the dominance of the convection. The gradient recovery–based adaptive mesh refinement strategy enables the method to provide high‐resolution numerical solutions by relatively fewer degrees of freedom. Moreover, the implementation detail of a surface mesh refinement technique is presented. Various numerical examples, including the convection‐dominated diffusion problems with large variations of solutions, nearly singular solutions, discontinuous sources, and internal layers on surfaces, are presented to demonstrate the efficacy and accuracy of the proposed method.
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