A stabilized finite element method for advection-diffusion equations on surfaces

Olshanskii, Maxim A.; Reusken, Arnold; Xu, Xianmin

doi:10.1093/imanum/drt016

Cited by 60 publications

(52 citation statements)

References 36 publications

(49 reference statements)

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“…Stabilization for advection-diffusion applications on manifolds were considered in the work of Olshanskii and Reusken. 34 The numerical results show that higher-order convergence rates are achieved provided that the finite element spaces are properly chosen. In addition, the conditioning of the system of equations depends on the element orders employed for the approximation of the individual physical fields.…”

Section: Introductionmentioning

confidence: 92%

Higher‐order surface FEM for incompressible Navier‐Stokes flows on manifolds

Fries

2018

Numerical Methods in Fluids

101

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Summary Stationary and instationary Stokes and Navier‐Stokes flows are considered on two‐dimensional manifolds, ie, on curved surfaces in three dimensions. The higher‐order surface FEM is used for the approximation of the geometry, velocities, pressure, and Lagrange multiplier to enforce tangential velocities. Individual element orders are employed for these various fields. Streamline‐upwind stabilization is employed for flows at high Reynolds numbers. Applications are presented, which extend classical benchmark test cases from flat domains to general manifolds. Highly accurate solutions are obtained, and higher‐order convergence rates are confirmed.

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Section: Introductionmentioning

confidence: 92%

Higher‐order surface FEM for incompressible Navier‐Stokes flows on manifolds

Fries

2018

Numerical Methods in Fluids

101

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“…Some different choices can be found in other works. 6,24,29,33,46,47 The stabilization parameter used in this paper is a combination of the strategies from the works of Elman et al 29 and Roos et al, 33 ie, using the gradually changing parameter for different regions according to their local mesh Peclet numbers.…”

Section: The Sdmmentioning

confidence: 99%

“…Convection-diffusion-reaction partial differential equations (PDEs) on surfaces are intensively applied to model the transport or migration processes in physical and biological science. Well-known application examples are the transport of surfactants on moving interfaces with application to multiphase flows [1][2][3][4][5][6] and the chemotaxis processes in which single colonies of bacteria cells merge together to form one larger colony on solid surfaces and biological films. [7][8][9][10] Surface convection and diffusion problems are also fundamental subproblems involved in the surface flow models with applications to geoscience.…”

Section: Introductionmentioning

confidence: 99%

“…Another popular finite element approach is called the volume mesh-based finite element method. 6,22 Its implementation is based on the outer tetrahedral mesh, which is different from the SFEM, but they have similar frameworks of theoretical analysis. The volume mesh-based finite element method is designed for problems in which there is a coupling of the surface flow and a flow from the outer domain.…”

Section: Introductionmentioning

confidence: 99%

“…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. Moreover, in the work of Olshanskii et al, 6 a streamline diffusion method (SDM) is proposed to overcome the instability issue. This method is improved by combining the adaptive mesh refinement (AMR) strategy in the work of Chernyshenko and Olshanskii.…”

Section: Introductionmentioning

confidence: 99%

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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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Simultaneous analysis of continuously embedded Reissner–Mindlin shells in 3D bulk domains

Kaiser,

Fries

2024

Numerical Meth Engineering

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A mechanical model and numerical method for the simultaneous analysis of Reissner–Mindlin shells with geometries implied by a continuous set of level sets (isosurfaces) over some three‐dimensional bulk domain is presented. A three‐dimensional mesh in the bulk domain is used in a tailored FEM formulation where the elements are by no means conforming to the level sets representing the shape of the individual shells. However, the shell geometries are bounded by the intersection curves of the level sets with the boundary of the bulk domain so that the boundaries are meshed conformingly. This results in a method which was coined Bulk Trace FEM before. The simultaneously considered, continuously embedded shells may be useful in the structural design process or for the continuous reinforcement of bulk domains. Numerical results confirm higher‐order convergence rates.

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A stabilized finite element method for advection-diffusion equations on surfaces

Cited by 60 publications

References 36 publications

Higher‐order surface FEM for incompressible Navier‐Stokes flows on manifolds

Higher‐order surface FEM for incompressible Navier‐Stokes flows on manifolds

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

Simultaneous analysis of continuously embedded Reissner–Mindlin shells in 3D bulk domains

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