Boundary methods for solving elliptic problems with singularities and interfaces
SIAM Journal on Numerical Analysis
Computational Mathematics and Mathematical Physics
A Finite-Element Method for Laplace- and Helmholtz-Type Boundary Value Problems with Singularities
SIAM Journal on Numerical Analysis
The High Accurate Block-Grid Method for Solving Laplace's Boundary Value Problem with Singularities
SIAM Journal on Numerical Analysis
A Singular Function Boundary Integral Method for Laplacian Problems with Boundary Singularities
SIAM Journal on Scientific Computing
Singularities and treatments of elliptic boundary value problems
Mathematical and Computer Modelling: An International Journal
Journal of Computational and Applied Mathematics
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The highly accurate block-grid method for solving Laplace's boundary value problems on polygons is developed for nonanalytic boundary conditions of the first kind. The quadrature approximation of the integral representations of the exact solution around each reentrant corner(''singular'' part) are combined with the 9-point finite difference equations on the ''nonsingular'' part. In the integral representations, and in the construction of the sixth order gluing operator, the boundary conditions are taken into account with the help of integrals of Poisson type for a half-plane which are computed with @e accuracy. It is proved that the uniform error of the approximate solution is of order O(h^6+@e), where h is the mesh step. This estimation is true for the coefficients of singular terms also. The errors of p-order derivatives (p=0,1,...) in the ''singular'' parts are O((h^6+@e)r"j^1^/^@a^"^j^-^p), r"j is the distance from the current point to the vertex in question and @a"j@p is the value of the interior angle of the jth vertex. Finally, we give the numerical justifications of the obtained theoretical results.