Numerical methods for the fractional Laplacian and related nonlocal PDEs

Thesis

The thesis develops and analyses finite element methods for the fractional Laplacian and related partial differential equations. Our work spans rigorous numerical analysis and scientific computing, addressing evolutionary diffusion problems and elliptic boundary value problems, studying numerical methods for the fractional Laplacian on bounded domains in both its spectral and integral definition.

A central contribution of the initial part of the thesis is the construction of the first provably convergent finite element scheme on a bounded domain for a parabolic regularisation of the porous medium equation with a fractional pressure. The model consists of a nonlinear diffusion equation in which the pressure is determined through the spectral fractional Laplacian with a Neumann boundary condition. The convergence proof also constitutes a constructive existence argument for weak solutions, while the scheme preserves mass conservation, nonnegativity of the density, and the free energy dissipation structure of the continuous problem. The analysis is then extended to a broader class of porous medium equations involving spectral fractional powers of second order elliptic operators, incorporating anisotropy and confinement. On the computational side, the thesis develops an efficient algorithm for the spectral fractional Laplacian based on rational approximation theory, circumventing the high computational cost of full spectral decompositions. This is used to perform the first study of the fractional Keller–Segel equation on bounded domains from a computing perspective, exploring blow-up phenomena and their dependence on mass and concentration of the initial datum.

We then study the integral fractional Laplacian on a bounded domain and a classical elliptic problem associated with it. In this context, the thesis proposes a finite element method with a weighted basis that exploits the known low regularity of solutions near the boundary, by incorporating this directly into the scheme, through a finite element basis having the same (low) regularity. Under natural regularity assumptions on the domain, the method is shown to improve all previously known error estimates in the fractional Sobolev norm. A companion quadrature rule for the resulting nonlocal bilinear form is also developed. The quadrature, efficiently computable through the use of the Riemann and Epstein zeta functions, enables a systematic study of the interplay between finite element discretization and bilinear form approximation for the finite element method with a weighted basis and comparison with the standard piecewise linear basis.

Taken together, the thesis makes original contributions to the rigorous numerical analysis of nonlocal PDEs, to the design of practically implementable algorithms, and to the mathematical understanding of fractional nonlocal diffusive processes on bounded domains.

Authors: Fronzoni, S

• DOI: 10.5287/ora-bmgzvpnwj
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: porous medium equation; fractional Poisson equation; fractional Laplacian; variational crime; nonlocal diffusion; Keller-Segel equation; finite element method; rational approximation; weighted finite element method
• Subjects: Numerical Analysis of PDEs; Mathematics