A posteriori error analysis for a cut cell finite volume method

Estep, D.; Pernice, Michael; Tavener, Simon; Wang, Haiying

doi:10.1016/j.cma.2010.11.015

Cited by 6 publications

(2 citation statements)

References 30 publications

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“…We consider the three dimensional case as future work. For other works treating a posteriori error estimation and cut cell techniques we refer to [27], where a finite volume method was considered, and [40] and [23] where cut cell methods were applied. This paper is organized as follows.…”

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confidence: 99%

Flux recovery for Cut Finite Element Method and its application in a posteriori error estimation

Capatina

2021

ESAIM: M2AN

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In this article, we aim to recover locally conservative and $H(div)$ conforming fluxes for the linear Cut Finite Element Solution with Nitsche's method for Poisson problems with Dirichlet boundary condition. The computation of the conservative flux in the Raviart-Thomas space is completely local and does not require to solve any mixed problem. The $L^2$-norm of the difference between the numerical flux and the recovered flux can then be used as a posteriori error estimator in the adaptive mesh refinement procedure. Theoretically we also prove the global reliability and local efficiency. The theoretical results are verified in the numerical results. Moreover, in the numerical results we also observe optimal convergence rate for the flux error.

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confidence: 99%

Flux recovery for Cut Finite Element Method and its application in a posteriori error estimation

Capatina

2021

ESAIM: M2AN

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“…Although we herein restrict the discussion to the case of piecewise affine approximation spaces from [15], we believe that the ideas introduced can be extended for instance to the high order case discussed in [16] as regards the CutFEM method, and in [7] as regards the flux reconstruction in two dimensions. For other works treating a posteriori error estimation and cut cell techniques we refer to [26], where a finite volume method was considered, and [39] and [22] where cut cell methods were applied.…”

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confidence: 99%

Flux recovery for Cut finite element method and its application in a posteriori error estimation

Capatina¹,

He²

2021

Preprint

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In this article, we aim to recover locally conservative and H(div) conforming fluxes for the linear Cut Finite Element Solution with Nitsche's method for Poisson problems with Dirichlet boundary condition. The computation of the conservative flux in the Raviart-Thomas space is completely local and does not require to solve any mixed problem. The L 2 -norm of the difference between the numerical flux and the recovered flux can then be used as a posteriori error estimator in the adaptive mesh refinement procedure. Theoretically we are able to prove the global reliability and local efficiency. The theoretical results are verified in the numerical results. Moreover, in the numerical results we also observe optimal convergence rate for the flux error.

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Reliable Residual-Based Error Estimation for the Finite Cell Method

Stolfo

Schröder

2021

J Sci Comput

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In this work, the reliability of a residual-based error estimator for the Finite Cell method is established. The error estimator is suitable for the application of hp-adaptive finite elements and allows for Neumann boundary conditions on curved boundaries. The reliability proof of the error estimator relies on standard arguments of residual-based a posteriori error control, but includes several modifications with respect to the error contributions associated with the volume residuals as well as the jumps across inner edges and Neumann boundary parts. Important ingredients of the proof are Stein’s extension theorem and a modified trace theorem which estimates the norm of the trace on (curved) boundary parts in terms of the local mesh size and polynomial degree. The efficiency of the error estimator is also considered by discussing an artificial example which yields an efficiency index depending on the mesh-family parameter h. Numerical experiments on more realistic domains, however, suggest global efficiency with the occurrence of a large overestimation on only few cut elements. In the experiments the reliability of the error estimator is demonstrated for h- and p-uniform as well as for hp-geometric and h-adaptive refinements driven by the error estimator. The practical applicability of the error estimator is also studied for a 3D problem with a non-smooth solution.

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A posteriori error analysis for a cut cell finite volume method

Cited by 6 publications

References 30 publications

Flux recovery for Cut Finite Element Method and its application in a posteriori error estimation

Flux recovery for Cut Finite Element Method and its application in a posteriori error estimation

Flux recovery for Cut finite element method and its application in a posteriori error estimation

Reliable Residual-Based Error Estimation for the Finite Cell Method

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