An FDM-sFEM scheme on time-space manifolds and its superconvergence analysis

TL;DR

This paper introduces a superconvergent discretization scheme for the Laplace-Beltrami operator on time-space manifolds, combining finite differences and surface finite elements, with rigorous error analysis and numerical validation.

Contribution

It presents a new coupled FDM-sFEM scheme with superconvergence analysis and a novel geometric error framework for dynamic surface problems.

Findings

Superconvergence of the gradient via post-processing techniques.

Validation of theoretical results through numerical examples.

A new summation by parts formula for analysis.

Abstract

We study superconvergent discretization of the Laplace-Beltrami operator on time-space product manifolds with Neumann temporal boundary values, which arise in the context of dynamic optimal transport on general surfaces. We propose a coupled scheme that combines finite difference methods in time with surface finite element methods in space. By establishing a new summation by parts formula and proving the supercloseness of the semi-discrete solution, we derive superconvergence results for the recovered gradient via post-processing techniques. In addition, our geometric error analysis is implemented within a novel framework based on the approximation of the Riemannian metric. Several numerical examples are provided to validate and illustrate the theoretical results.