A gradient recovery–based adaptive finite element method for convection‐diffusion‐reaction equations on surfaces

Abstract

In 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. KEYWORDSadaptive strategy, recovery-based error estimator, streamline diffusion method, surface convection-diffusion-reaction equations, surface finite element method MSC CLASSIFICATION 65N30, 65N12, 65N50, 58J32, 76R05

Modified Characteristic Finite Element Method with Second-Order Spatial Accuracy for Solving Convection-Dominated Problem on Surfaces

“…The numerical methods for solving partial differential Equations (PDEs) on surfaces can be divided into two main categories: mesh-free methods [8][9][10][11][12] and mesh-based methods [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. For mesh-free methods, the implementation of this particular method is relatively straightforward.…”

“…We focus here on the finite element method which is one of the mesh-based methods for solving PDEs on surfaces. The mesh generation include two prevalent strategies: embedding the surface in the narrow-band domain [16][17][18][19][20][21][22][23] and directly discretizing the surface [25][26][27][28][29][30].…”

“…For the same problem, Simon and Tobiska provide a priori error estimation for a fitted finite element local projection stabilization in their study, which is symmetric stability terms [26]. Recently, Xiao et al developed gradient recovery-based adaptive techniques in [27]. Jin et al [28] subsequently utilized their work [27] to address the convection-dominated problem on surfaces using mixed finite element methods and provided theoretical analysis.…”

“…Recently, Xiao et al developed gradient recovery-based adaptive techniques in [27]. Jin et al [28] subsequently utilized their work [27] to address the convection-dominated problem on surfaces using mixed finite element methods and provided theoretical analysis.…”

An enriched Galerkin-characteristics finite element method for convection-dominated and transport problems

“…This second approach is the most popular strategy of evaluating error behavior and it is adopted in the present work. In contrast to the gradient-based h-adaptive finite element methods as those investigated in [23,24,25], linear systems in the proposed enriched Galerkin-characteristics finite element method keep the same structure and size at each adaptation step. Indeed, for the gradient-based h-adaptive methods, an initial coarse mesh is needed to compute a primary solution for evaluating the gradient.…”