A parametrix for the surface Stokes equation

TL;DR

This paper presents a new integral equation formulation for the surface Stokes equations using Stokeslets, along with a fast solver based on a proxy shell method, enabling high-order discretization and efficient computation.

Contribution

It introduces a novel integral equation approach for surface Stokes equations and develops a fast direct solver using a proxy shell method.

Findings

The integral equations are of the second kind and suitable for high-order discretization.

The proxy shell method effectively constructs fast solvers for dense linear systems.

Numerical examples demonstrate the accuracy and efficiency of the proposed scheme.

Abstract

We introduce an integral equation formulation of the surface Stokes equations, constructed using two-dimensional Stokeslets. The resulting integral equations are Fredholm integral equations of the second kind and can be discretized to high order using standard tools. Since the resulting discrete linear systems are dense, we describe and analyze a proxy shell method to construct fast direct solvers for these systems. The properties of our integral equation, and the performance of the resulting numerical scheme, are illustrated with several representative numerical examples.