Efficient finite element methods for the integral fractional Laplacian and applications

Abstract

This thesis addresses the development of efficient finite element methods and their analysis for nonlocal problems, with particular focus on the integral fractional Laplacian. The specific topics addressed in this thesis are: regularity theory of solutions in polygonal domains, graded and hp versions of the finite element method, operator preconditioning, space–time adaptive methods for variational inequalities, and interface problems. The numerical analysis is supplemented by applications to biological and robotic systems. The precise regularity theory of the solutions in polygonal domains is first addressed as a basis for the numerical analysis. It is shown that the solution admits an asymptotic expansion with a tensor product decomposition, which leads to the optimal rate of convergence for finite element discretisations on graded meshes and for the hp–version on quasi–uniform meshes. An operator preconditioner for general elliptic pseudodifferential equations in a domain is then presented, based on a classical formula by Boggio. The thesis also considers the a priori and a posteriori analysis of a large class of space and space–time variational inequalities associated with the fractional Laplacian. The resulting space–time adaptive methods are studied in numerical experiments. Two further chapters of this thesis study applications to biological and robotic systems. Analysis and numerical experiments of the resulting continuum nonlocal equations allow for efficient quantitative characterisation of relevant quantities.