# Boundary Integral Analysis for the Non-homogeneous 3D Stokes Equation

## TL;DR

This paper introduces a boundary integral method for efficiently solving the non-homogeneous 3D Stokes equation by transforming volume integrals into boundary integrals, validated through numerical tests.

## Contribution

It presents a novel boundary integral reformulation for the 3D Stokes equation that simplifies volume integrals into boundary integrals, enabling efficient numerical computation.

## Key findings

- The method successfully reduces volume integrals to boundary integrals.
- Numerical validation confirms the accuracy of the boundary integral approach.
- The approach facilitates potential extensions to Navier-Stokes equations.

## Abstract

A regular-grid volume-integration algorithm is developed for the non-homogeneous 3D Stokes equation. Based upon the observation that the Stokeslet ${\mathcal U}$ is the Laplacian of a function ${\mathcal H}$, the volume integral is reformulated as a simple boundary integral, plus a remainder domain integral. The modified source term in this remainder integral is everywhere zero on the boundary and can therefore be continuously extended as zero to a regular grid covering the domain. The volume integral can then be evaluated on the grid. Applying this method to the Navier-Stokes equations will require obtaining velocity gradients, and thus an efficient algorithm for post-processing these derivatives is also discussed. To validate the numerical implementation, test results employing a linear element Galerkin approximation are presented.