Convergence analysis of three semi-discrete numerical schemes for nonlocal geometric flows including perimeter terms

TL;DR

This paper analyzes three semi-discrete numerical schemes for nonlocal geometric flows with perimeter terms, providing convergence proofs and numerical validation for each scheme's accuracy under various norms.

Contribution

It introduces and rigorously analyzes three distinct semi-discrete schemes for nonlocal geometric flows, establishing their convergence rates and error estimates.

Findings

Quadratic convergence of the finite difference scheme under $H^1$-norm.

Linear convergence of finite element schemes under $H^1$-norm.

Numerical experiments confirm theoretical convergence and accuracy.

Abstract

We present and analyze three distinct semi-discrete schemes for solving nonlocal geometric flows incorporating perimeter terms. These schemes are based on the finite difference method, the finite element method, and the finite element method with a specific tangential motion. We offer rigorous proofs of quadratic convergence under $H^1$-norm for the first scheme and linear convergence under $H^1$-norm for the latter two schemes. All error estimates rely on the observation that the error of the nonlocal term can be controlled by the error of the local term. Furthermore, we explore the relationship between the convergence under $L^\infty$-norm and manifold distance. Extensive numerical experiments are conducted to verify the convergence analysis, and demonstrate the accuracy of our schemes under various norms for different types of nonlocal flows.