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Author: Heida, Martin × End date: 2023 × Has files: Yes × Type: Article × Version: publishedVersion × WGL Institute: WIAS ×

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    Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator
    (Les Ulis : EDP Sciences, 2021) Heida, Martin; Kantner, Markus; Stephan, Artur
    We introduce a family of various finite volume discretization schemes for the Fokker–Planck operator, which are characterized by different Stolarsky 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 choice of the Stolarsky weights. We show that the Scharfetter–Gummel scheme has the analytically best convergence properties but also that there exists a whole branch of Stolarsky means with the same convergence quality. We show by numerical experiments that for small convection 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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    Stochastic homogenization on perforated domains II – Application to nonlinear elasticity models
    (Berlin : Wiley-VCH, 2022) Heida, Martin
    Based on a recent work that exposed the lack of uniformly bounded (Formula presented.) extension operators on randomly perforated domains, we study stochastic homogenization of nonlinear p-elasticity, (Formula presented.), on such structures using instead the extension operators constructed in former works. We thereby introduce two-scale convergence methods on such random domains under the intrinsic loss of regularity and prove some generally useful calculus theorems on the probability space, for example, abstract Gauss theorems.