The Block-Grid Method for Solving Laplace's Equation on Polygons with Nonanalytic Boundary Conditions

Abstract

The block-grid method see Dosiyev, 2004 for the solution of the Dirichlet problem on polygons, when a boundary function on each side of the boundary is given from C 2,λ , 0 < λ < 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 h 2 ε , where ε is the error of the approximation of the mentioned integrals, h is the mesh step. For the p-order derivatives p 0, 1, . . . of the difference between the approximate and the exact solution in each "singular" part O h 2 ε r 1/αj −p j order is obtained, here r j is the distance from the current point to the vertex in question, α j π is the value of the interior angle of the jth 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.

Numerical Solution of Sylvester Equation Based on Iterative Predictor‐Corrector Method

“…It can also be useful when Λ is rectangular matrix or an ill-conditioned matrix. e predictor-corrector iterative method may also be applied to precondition the coefficient matrices of the linear algebraic system of equations obtained by using finite difference method to solve boundary value problems, for example, for the solution of Laplace's equation with singularities (see [26,27]), for the solution of the heat equation on hexagonal grid (see [28]), and for the approximation of the derivatives of the solution of the heat equation (see [29,30]). Moreover, the iterative method of predictor-corrector for finding matrix inverses may be applied to precondition the coefficient matrices of the system of equations obtained by using finite element method to solve parabolic partial differential equations (see [31]).…”

One-block method for computing the generalized stress intensity factors for Laplace’s equation on a square with a slit and on an L-shaped domain

“…These coefficients have great importance in many applications, especially in fracture mechanics. The methods for the approximation of the coefficients a j can be divided into two groups: (i) the post-processing approach in the finite element or finite difference methods (see [5][6][7][8][9][10]), in the block method (see [11][12][13]) and in the block-grid method [14][15][16] (ii) directly calculated methods, without finding approximate solution of the boundary value problem (see [4,1,[17][18][19][20] and the references therein).…”

A Fourth-Order Block-Grid Method for Solving Laplace's Equation on a Staircase Polygon with Boundary Functions in

“…In [8,9], the restriction on the boundary functions to be algebraic polynomials on the sides of the polygon causing the singular vertices in the BGM was removed. It was assumed that the boundary function on each side of the polygon is given from the Hölder classes , , 0 < < 1, and on the "nonsingular" part the 5-point scheme is used when = 2 [8] and the 9-point scheme is used when = 6 [9]. For the 5-point scheme a simple linear interpolation with 4 points is used.…”