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An adapted Galerkin method for the resolution of Dirichlet and Neumann problems in a polygonal domain
Author(s) -
Bourlard Maryse,
Nicaise Serge,
Paquet Luc
Publication year - 1990
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670120307
Subject(s) - mathematics , galerkin method , mathematical analysis , dirichlet problem , neumann boundary condition , laplace's equation , integral equation , fredholm integral equation , dirichlet boundary condition , laplace transform , boundary value problem , domain (mathematical analysis) , finite element method , physics , thermodynamics
The Neumann problem for Laplace's equation in a polygonal domain is associated with the exterior Dirichlet problem obtained by requiring the continuity of the potential through the boundary. Then the solution is the simple layer potential of the charge q on the boundary. q is the solution of a Fredholm integral equation of the second kind that we solve by the Galerkin method. The charge q has a singular part due to the corners, so the optimal order of convergence is not reached with a uniform mesh. We restore this optimal order by grading the mesh adequately near the corners. The interior Dirichlet problem is solved analogously, by expressing the solution as a double layer potential.