Local projection stabilization for convection–diffusion–reaction equations on surfaces

Simon, Kristin; Tobiska, Lutz

doi:10.1016/j.cma.2018.09.031

Cited by 6 publications

(5 citation statements)

References 14 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

“…However, the accuracy and stability of this method depend on the careful selection of the stabilization parameters, which requires rigorous theoretical considerations. 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].…”

Section: Introductionmentioning

confidence: 99%

“…The characteristic finite element method [23, 29,31] is easier to operate than the above methods [22,[26][27][28]. To implement this method, it is necessary to convert the CRD equation into a reaction-diffusion equation and interpolate the solution in the characteristic direction [31].…”

Section: Introductionmentioning

confidence: 99%

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

Wu,

Feng,

2023

Entropy

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We present a modified characteristic finite element method that exhibits second-order spatial accuracy for solving convection–reaction–diffusion equations on surfaces. The temporal direction adopted the backward-Euler method, while the spatial direction employed the surface finite element method. In contrast to regular domains, it is observed that the point in the characteristic direction traverses the surface only once within a brief time. Thus, good approximation of the solution in the characteristic direction holds significant importance for the numerical scheme. In this regard, Taylor expansion is employed to reconstruct the solution beyond the surface in the characteristic direction. The stability of our scheme is then proved. A comparison is carried out with an existing characteristic finite element method based on face mesh. Numerical examples are provided to validate the effectiveness of our proposed method.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Wu,

Feng,

2023

Entropy

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“…For the methods based on the SFEM, a local projection stabilization is extended to the SFEM for solving the convection-diffusion-reaction equations on surfaces. 23 In addition, two types of spurious oscillations at layers diminishing methods, ie, an edge stabilization and the Mizukami-Hughes Petrov-Galerkin methods, are provided to eliminate the nonphysical oscillations on surfaces. 24 For the methods based on the volume mesh-based finite element method, an edge stabilization method is studied in the work of Burman et al 25 for the convection-dominated problems on surfaces.…”

Section: Introductionmentioning

confidence: 99%

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

Xiao

Feng

2019

Numerical Meth Engineering

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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

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A cubic B-spline semi-analytical algorithm for simulation of 3D steady-state convection-diffusion-reaction problems

Lin

Reutskiy

2020

Applied Mathematics and Computation

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Local projection stabilization for convection–diffusion–reaction equations on surfaces

Cited by 6 publications

References 14 publications

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

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

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

A cubic B-spline semi-analytical algorithm for simulation of 3D steady-state convection-diffusion-reaction problems

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