Analysis of the Block-Grid Method for the Solution of Laplace's Equation on Polygons with a Slit

The error estimates obtained for solving Laplace's boundary value problem on polygons by the block-grid method contain constants that are difficult to calculate accurately. Therefore, the experimental analysis of the method could be essential. The real characteristics of the block-grid method for solving Laplace's equation on polygons with a slit are analysed by experimental investigations. The numerical results obtained show that the order of convergence of the approximate solution is the same as in the case of a smooth solution. To illustrate the singular behaviour around the singular point, the shape of the highly accurate approximate solution and the figures of its partial derivatives up to second order are given in the “singular” part of the domain. Finally a highly accurate formula is given to calculate the stress intensity factor, which is an important quantity in fracture mechanics

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

The integral representations of the solution around the vertices of the interior reentered angles (on the “singular” parts) are approximated by the composite midpoint rule when the boundary functions are from These approximations are connected with the 9-point approximation of Laplace's equation on each rectangular grid on the “nonsingular” part of the polygon by the fourth-order gluing operator. It is proved that the uniform error is of order where and is the mesh step. For the -order derivatives () of the difference between the approximate and the exact solutions, in each “ singular” part order is obtained; here is the distance from the current point to the vertex in question and is the value of the interior angle of the th vertex. Numerical results are given in the last section to support the theoretical results

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 , , 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 , where is the error of the approximation of the mentioned integrals, is the mesh step. For the -order derivatives ( ) of the difference between the approximate and the exact solution in each "singular" part order is obtained, here is the distance from the current point to the vertex in question, is the value of the interior angle of the 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.</p