# Integral equation formulation of the biharmonic Dirichlet problem

## TL;DR

This paper introduces a new integral representation for solving the biharmonic Dirichlet problem, improving robustness and accuracy, especially on high-curvature domains, by leveraging a connection to Stokes flow.

## Contribution

The authors develop a novel integral formulation for the biharmonic Dirichlet problem using a Stokes problem analogy, applicable to complex domains, enhancing computational stability.

## Key findings

- New integral representation with better kernel behavior on high-curvature domains
- Augmentation techniques for handling simply and multiply connected domains
- Numerical examples demonstrating improved robustness and accuracy

## Abstract

We present a novel integral representation for the biharmonic Dirichlet problem. To obtain the representation, the Dirichlet problem is first converted into a related Stokes problem for which the Sherman-Lauricella integral representation can be used. Not all potentials for the Dirichlet problem correspond to a potential for Stokes flow, and vice-versa, but we show that the integral representation can be augmented and modified to handle either simply or multiply connected domains. The resulting integral representation has a kernel which behaves better on domains with high curvature than existing representations. Thus, this representation results in more robust computational methods for the solution of the Dirichlet problem of the biharmonic equation and we demonstrate this with several numerical examples.