Fractal homogenization of multiscale interface problems

Heida, Martin; Kornhuber, Ralf; Podlesny, Joscha

doi:10.48550/arxiv.1712.01172

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Numerical Simulation and Convergence Analysis1

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2020

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“…The idea behind the proof seems to go back to Hummel [Hum99] and has been adapted in a series of works e.g. [Hei18,HKP17]. Let e 0 = 0 and (e i ) i=1,...,n be the canonical basis of R n .…”

Section: Numerical Simulation and Convergence Analysismentioning

confidence: 99%

Consistency and convergence for a family of finite volume discretizations of the Fokker--Planck operator

Heida¹,

Kantner²,

Stephan³

2020

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We introduce a family of various finite volume discretization schemes for the Fokker-Planck operator, which are characterized by different weight functions on the edges. This family particularly includes the well-established Scharfetter-Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the quality of the weights. We show by numerical experiments that for small gradients the choice of the optimal representative of the discretization family is highly non-trivial while for large gradients the Scharfetter-Gummel scheme stands out compared to the others.

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“…The idea behind the proof seems to go back to Hummel [Hum99] and has been adapted in a series of works e.g. [Hei18,HKP17]. Let e 0 = 0 and (e i ) i=1,...,n be the canonical basis of R n .…”

Section: Numerical Simulation and Convergence Analysismentioning

confidence: 99%