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

Fries, Thomas‐Peter

doi:10.1002/fld.4510

Cited by 66 publications

(103 citation statements)

References 51 publications

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“…2(e) shows the resulting flow field for an instationary Navier-Stokes flow and the resulting Kármán vortex street can clearly be seen. The test cases are documented in detail in [4,5].…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The boundary conditions extend in time and a divergence-free, tangential, initial condition for u is needed. The (stabilized) weak forms of the governing equations are found in [4,5] and are omitted here for brevity. The surface FEM [6] is used for the discretization.…”

Section: Governing Equationsmentioning

confidence: 99%

“…Some benchmark test cases are suggested and highly accurate solutions obtained. This work is a short summary of [4,5].…”

Section: Introductionmentioning

confidence: 99%

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Higher‐order surface FEM for incompr. Navier‐Stokes flows on manifolds

Fries

2018

Proc Appl Math and Mech

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Stationary and instationary Stokes and Navier-Stokes flows are considered on curved surfaces in three dimensions. A stabilized, mixed, higher-order surface FEM with individual element orders is used for the approximation of the geometry, velocities, pressure, and Lagrange multiplier to enforce tangential velocities. Benchmark test cases are proposed and higher-order convergence rates are confirmed.

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“…2(e) shows the resulting flow field for an instationary Navier-Stokes flow and the resulting Kármán vortex street can clearly be seen. The test cases are documented in detail in [4,5].…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: Governing Equationsmentioning

confidence: 99%

See 1 more Smart Citation

Higher‐order surface FEM for incompr. Navier‐Stokes flows on manifolds

Fries

2018

Proc Appl Math and Mech

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“…[7][8][9][10] Surface convection and diffusion problems are also fundamental subproblems involved in the surface flow models with applications to geoscience. [11][12][13][14][15][16] While there are a lot of discussions on numerical methods in the literature for convection-diffusion-reaction PDEs in one, two, and three dimensions, it is challenging to develop efficient numerical methods for such PDEs on surfaces (or manifolds).…”

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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“…The P 2 -P 1 continuous Taylor-Hood element is one of the most popular FE pairs for incompressible fluid flow problems. Surface variants of this pair have been used for surface Navier-Stokes equations in the recent papers [35,12]. In those papers the fitted surface FE approach is used.…”

mentioning

confidence: 99%

Inf-sup stability of the trace $\mathbf {P}_2$–$P_1$ Taylor–Hood elements for surface PDEs

2021

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The paper studies a geometrically unfitted finite element method (FEM), known as trace FEM or cut FEM, for the numerical solution of the Stokes system posed on a closed smooth surface. A trace FEM based on standard Taylor-Hood (continuous P 2 -P 1 ) bulk elements is proposed. A so-called volume normal derivative stabilization, known from the literature on trace FEM, is an essential ingredient of this method. The key result proved in the paper is an inf-sup stability of the trace P 2 -P 1 finite element pair, with the stability constant uniformly bounded with respect to the discretization parameter and the position of the surface in the bulk mesh. Optimal order convergence of a consistent variant of the finite element method follows from this new stability result and interpolation properties of the trace FEM. Properties of the method are illustrated with numerical examples.

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

Cited by 66 publications

References 51 publications

Higher‐order surface FEM for incompr. Navier‐Stokes flows on manifolds

Higher‐order surface FEM for incompr. Navier‐Stokes flows on manifolds

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

Inf-sup stability of the trace $\mathbf {P}_2$–$P_1$ Taylor–Hood elements for surface PDEs

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