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In Boundary Element Method, Green's function with no boundary conditions isused for solving Laplace's equation with Dirichlet boundary condition. Todetermine the gradient of solution on the boundary, we need to solve theboundary integral equation numerically in most practical cases. Here we discussthe alternative method to avert solving that boundary integral equation. It isbased on the solution for Poisson's equation which has the singularity on theboundary specified by the boundary function and shown it is applicable toarbitrary shaped domain and reduces calculation cost considerably.

A solution of Dirichlet problem using second partial derivatives of boundary function