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The block-grid method for solving Laplace’s equation on polygons with nonanalytic boundary conditions. (English) Zbl 1214.65054
Summary: The block-grid method [A. A. Dosiev, Comput. Math. Math. Phys. 34, No. 5, 591–604 (1994; Zbl 0832.65113); translation from Zh. Vychisl. Mat. Mat. Fiz. 34, No. 5, 685–701 (1994)] for the solution of the Dirichlet problem on polygons, when a boundary function on each side of the boundary is given from ${C}^{2,\lambda },0<\lambda <1$, is analized. In the integral represetations around each singular vertex, which are combined with the uniform grids on “nonsingular" part the boundary conditions are taken into account with the help of integrals of Poisson type for a half-plane. It is proved that the final uniform error is of order $O\left({h}^{2}+ϵ\right)$, where $\epsilon$ is the error of the approximation of the mentioned integrals, $h$ is the mesh step. For the $p$-order derivatives ($p=0,1,\cdots$) of the difference between the approximate and the exact solution in each “singular" part $O\left(\left({h}^{2}+\epsilon \right){r}_{j}^{1/{\alpha }_{j}-p}\right)$ order is obtained, here ${r}_{j}$ is the distance from the current point to the vertex in question, ${\alpha }_{j}\pi$ is the value of the interior angle of the $j$th vertex. Finally, the method is illustrated by solving the problem in L-shaped polygon, and a high accurate approximation for the stress intensity factor is given.
##### MSC:
 65N06 Finite difference methods (BVP of PDE) 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation 35J25 Second order elliptic equations, boundary value problems