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Nonlocal models with truncated interaction kernels - analysis, finite element methods and shape optimization

  • Nonlocal operators are used in a wide variety of models and applications due to many natural phenomena being driven by nonlocal dynamics. Nonlocal operators are integral operators allowing for interactions between two distinct points in space. The nonlocal models investigated in this thesis involve kernels that are assumed to have a finite range of nonlocal interactions. Kernels of this type are used in nonlocal elasticity and convection-diffusion models as well as finance and image analysis. Also within the mathematical theory they arouse great interest, as they are asymptotically related to fractional and classical differential equations. The results in this thesis can be grouped according to the following three aspects: modeling and analysis, discretization and optimization. Mathematical models demonstrate their true usefulness when put into numerical practice. For computational purposes, it is important that the support of the kernel is clearly determined. Therefore nonlocal interactions are typically assumed to occur within an Euclidean ball of finite radius. In this thesis we consider more general interaction sets including norm induced balls as special cases and extend established results about well-posedness and asymptotic limits. The discretization of integral equations is a challenging endeavor. Especially kernels which are truncated by Euclidean balls require carefully designed quadrature rules for the implementation of efficient finite element codes. In this thesis we investigate the computational benefits of polyhedral interaction sets as well as geometrically approximated interaction sets. In addition to that we outline the computational advantages of sufficiently structured problem settings. Shape optimization methods have been proven useful for identifying interfaces in models governed by partial differential equations. Here we consider a class of shape optimization problems constrained by nonlocal equations which involve interface-dependent kernels. We derive the shape derivative associated to the nonlocal system model and solve the problem by established numerical techniques.

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Author:Christian Vollmann
Referee:Prof. Dr. Volker Schulz, Prof. Dr. Max Gunzburger, JProf. Dr. Martin Siebenborn
Advisor:Prof. Dr. Volker Schulz, Prof. Dr. Leonhard Frerick
Document Type:Doctoral Thesis
Date of completion:2019/08/16
Publishing institution:Universit├Ąt Trier
Granting institution:Universit├Ąt Trier, Fachbereich 4
Date of final exam:2019/07/26
Release Date:2019/08/27
Tag:Nonlocal convection-diffusion; finite element method; shape optimization; multilevel Toeplitz; fractional Poisson equation; local limit
Institutes:Fachbereich 4
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):License LogoCC BY-NC-ND: Creative-Commons-Lizenz 4.0 International

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