A numerical method to solve the Stokes problem with a punctual force in source term

TL;DR

This paper introduces a finite-element numerical method for solving the Stokes problem with a Dirac source term, leveraging a fundamental solution to maintain optimal approximation accuracy, relevant for modeling active thin structures in viscous fluids.

Contribution

It presents a novel finite-element approach that preserves optimality for any approximation order when solving Stokes problems with punctual forces.

Findings

Method achieves optimal accuracy for all approximation orders.

Applicable to modeling active thin structures in viscous fluids.

Provides a fundamental solution-based framework for punctually forced Stokes problems.

Abstract

The aim of this note is to present a numerical method to solve the Stokes problem in a bounded domain with a Dirac source term, which preserves optimality for any approximation order by the finite-element method. It is based on the knowledge of a fundamental solution to the associated operator over the whole space. This method is motivated by the modeling of the movement of active thin structures in a viscous fluid.