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Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator
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.
Precompact probability spaces in applied stochastic homogenization
We provide precompactness and metrizability of the probability space Ω for random measures and random coefficients such as they widely appear in stochastic homogenization and are typically given from data. We show that these properties are enough to implement the convenient two-scale formalism by Zhikov and Piatnitsky (2006). To further demonstrate the benefits of our approach we provide some useful trace and extension operators for Sobolev functions on Ω, which seem not known in literature. On the way we close some minor gaps in the Sobolev theory on Ω which seemingly have not been proven up to date.
Consistency and convergence for a family of finite volume discretizations of the Fokker--Planck operator
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.
Stochastic two-scale convergence and Young measures
In this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic two-scale convergence.
Stochastic homogenization on perforated domains II – Application to nonlinear elasticity models
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.
Stochastic homogenization on irregularly perforated domains
We study stochastic homogenization of a quasilinear parabolic PDE with nonlinear microscopic Robin conditions on a perforated domain. The focus of our work lies on the underlying geometry that does not allow standard homogenization techniques to be applied directly. Instead we prove homogenization on a regularized geometry and demonstrate afterwards that the form of the homogenized equation is independent from the regularization. Then we pass to the regularization limit to obtain the anticipated limit equation. Furthermore, we show that Boolean models of Poisson point processes are covered by our approach.
Consistency and order 1 convergence of cell-centered finite volume discretizations of degenerate elliptic problems in any space dimension
We study consistency of cell-centered finite difference methods for elliptic equations with degenerate coefficients in any space dimension $dgeq2$. This results in order of convergence estimates in the natural weighted energy norm and in the weighted discrete $L^2$-norm on admissible meshes. The cells of meshes under consideration may be very irregular in size. We particularly allow the size of certain cells to remain bounded from below even in the asymptotic limit. For uniform meshes we show that the order of convergence is at least 1 in the energy semi-norm, provided the discrete and continuous solutions exist and the continuous solution has $H^2$ regularity.
GENERIC for dissipative solids with bulk-interface interaction
The modeling framework of GENERIC was originally introduced by Grmela and Öttinger for thermodynamically closed systems. It is phrased with the aid of the energy and entropy as driving functionals for reversible and dissipative processes and suitable geometric structures. Based on the definition functional derivatives we propose a GENERIC framework for systems with bulk-interface interaction and apply it to discuss the GENERIC structure of models for delamination processes.
Stochastic homogenization on perforated domains II -- Application to nonlinear elasticity models
Based on a recent work that exposed the lack of uniformly bounded W1,p → W1,p extension operators on randomly perforated domains, we study stochastic homogenization of nonlinear elasticity on such structures using instead the extension operators constructed in [11]. 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 Ω, e.g. abstract Gauss theorems.
Homogenization of a porous intercalation electrode with phase separation
In this work, we derive a new model framework for a porous intercalation electrode with a phase separating active material upon lithium intercalation. We start from a microscopic model consisting of transport equations for lithium ions in an electrolyte phase and intercalated lithium in a solid active phase. Both are coupled through a Neumann--boundary condition modeling the lithium intercalation reaction. The active material phase is considered to be phase separating upon lithium intercalation. We assume that the porous material is a given periodic microstructure and perform analytical homogenization. Effectively, the microscopic model consists of a diffusion and a Cahn--Hilliard equation, whereas the limit model consists of a diffusion and an Allen--Cahn equation. Thus we observe a Cahn--Hilliard to Allen--Cahn transition during the upscaling process. In the sense of gradient flows, the transition goes in hand with a change in the underlying metric structure of the PDE system.