Extraction method for Stokes Flow with jumps in the pressure

TL;DR

This paper introduces a novel numerical scheme for stationary Stokes flow with pressure jumps along interfaces, enabling the use of uniform grids and achieving optimal convergence rates for pressure and velocity.

Contribution

A new finite element method that handles pressure jumps in Stokes flow by constructing and removing an approximate singular function, simplifying computations on uniform grids.

Findings

Achieves $O(h)$ convergence for pressure

Achieves $O(h^2)$ convergence for velocity

Enables uniform grid application in complex interface problems

Abstract

In this paper, we consider a stationary, constant viscosity, incompressible Stokes flow with singular forces along one or several interfaces. Assuming only the jumps of the pressure are present along the interface, we develop a new numerical scheme for such a problem. By constructing an approximate singular function and removing it, we can apply a standard finite element method to solve it. A main advantage of our scheme is that one can use a uniform grid. We observe optimal $O(h)$ order for the pressure and $O(h^2)$ order for the velocity.