Integral Equation Formulation of the Biharmonic Dirichlet Problem
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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.
KeywordsIntegral equations Biharmonic Dirichlet Multiply connected
Mathematics Subject Classification31A10 31A30 65R20 65N99
The authors would like to thank Leslie Greengard for suggesting this problem and both Leslie Greengard and Shidong Jiang for many useful discussions.
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