Integral equation formulation of the biharmonic Dirichlet problem
This provides a more robust integral representation for solving the biharmonic Dirichlet problem, benefiting computational methods in applied mathematics and engineering.
The authors present a novel integral representation for the biharmonic Dirichlet problem that handles simply and multiply connected domains, with a kernel that behaves better on high-curvature domains, leading to more robust computational methods demonstrated through numerical examples.
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.