A Note on Solving the Generalized Dirichlet to Neumann Map on Irregular Polygons using Generic Factored Approximate Sparse Inverses
E-N.G. Grylonakis, C.K. Filelis-Papadopoulos and G.A. Gravvanis

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 109, No. 6, pp. 505-517, 2015
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Keywords Laplace Equation, Dirichlet-Neumann map, Global Relation, Collocation, Generic Factored Approximate Sparse Inverses, Preconditioned Generalized Minimum Residual restarted method.

A new transform method for solving boundary value problems in two dimensions was proposed by A.S. Fokas, namely the unified transform. This approach seeks a solution to the unknown boundary values by solving a global relation, using the known boundary data. This relation can be used to characterize the Dirichlet to Neumann map. For the numerical solution of the global relation, a collocation-type method was recently introduced. Hence, the considered method is used for solving the 2D Laplace equation in several irregular convex polygons. The linear system, resulting from the collocation-type method, was solved by the Explicit Preconditioned Generalized Minimum Residual restarted method in conjunction with the Modified Generic Factored Approximate Sparse Inverse matrix. Numerical results indicating the applicability of the proposed preconditioning scheme are provided, along with discussions on the implementation details of the method.

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