Regularization and Multi-level Tools in the Method of Fundamental Solutions

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The Method of Fundamental Solution is applied to potential problems. The source and collocation points are supposed to coincide and are located along the boundary. The singularities due to the singularity of the fundamental solution are avoided by several techniques (regularization and desingularization). Both the monopole and the dipole formulations are investigated. The resulting algebraic systems have advantageous properties provided that pure Dirichlet or pure Neumann boundary condition is prescribed. Otherwise, the original problem is converted to a sequence of pure Dirichlet and pure Neumann subproblems, the solutions of which converge rapidly to the solution of the original mixed problem. The iteration is embedded to a multi-level context in a natural way. Thus, the computational cost can be significantly reduced, and the problem of large and ill-conditioned matrices is also avoided.