Stable and convergent numerical methods for geometric and physical PDEs

Abstract

The first part (Chapter 2 and 3) of this thesis discusses the stability issues arising from approximating geometric flows (mean curvature flow, Willmore flow, etc.) using finite element method of parametric type. This topic is rather important because the lack of numerical stability is the main obstacle in the way of getting convergence proofs. The contributions of this part (Chapter 2 and 3) are the first ever convergence proof with a L2-suboptimal error estimate for Dziuk’s method for mean curvature flow of closed surfaces (open problem since 1990) and the first ever convergence proof with a L2-optimal error estimate for a stabilized BGN method for mean curvature flow of closed curves (open problem since 2008). Dziuk’s and BGN method are the two most important algorithms in the field of numerical geometric flows. Alongside giving the first ever convergence proofs to these two methods, we also develop a new framework of analysing the general behaviours of the finite element approximation to geometric flows. This framework is expected to be a new powerful tool which would help to design and analyse robust and convergent algorithms where our design and analysis of a stabilized version of the BGN method in Chapter 3 is the first example of this kind. The methodologies and treatments developed in Chapter 2 and 3 are hopeful to become standard in the future. The second part (Chapter 4–8) is a collection of miscellaneous topics with focus on the design of robust algorithms, stability analysis and convergence proof of the numerical methods for various linear and nonlinear PDEs arsing in geometry and physics.