Laplace Boundary Value Research Articles

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 high accurate composite grid method for solving Laplace's boundary value problems with singularities

A high accurate composite grid method for solving Laplace's boundary value problems with singularities

The High Accurate Block-Grid Method for Solving Laplace's Boundary Value Problem with Singularities

A high accurate difference-analytical method is introduced for the solution of the mixed boundary value problem for Laplace's equation on graduated polygons. The polygon can have broken sections and be multiply connected. The uniform estimate of the error of the approximate solution is of order O(h6 ), whereas it is of order $O(h^{6}/r_{j}^{p-\lambda _{j}})$ for the errors of p-order derivatives (p=1,2,. . .) in a finite neighborhood of reentry vertices; here, h is the mesh step, rj is the distance from the current point to the vertex in question, $\lambda _{j}=1/(a\alpha _{j})$, and a=1 or 2 depending on the types of boundary conditions. Further, $\alpha _{j}\pi $ is the value of the interior angle at the considered vertex. Numerical experiments are illustrated in section \ref{sec8} to support the analysis made.

A nonconforming combined method for solving Laplace's boundary value problems with singularities

A nonconforming combined method for solving Laplace's boundary value problems with singularities

A nonconforming combined method for solving Laplace's boundary value problems with singularities

For solving Laplace's boundary value problems with singularities, a nonconforming combined approach of the Ritz-Galerkin method and the finite element method is presented. In this approach, singular functions are chosen to be admissible functions in the part of a solution domain where there exist singularities; and piecewise linear functions are chosen to be admissible functions in the rest of the solution domain. In addition, the admissible functions used here are constrained to be continuous only at the element nodes on the common boundary of both methods. This method is nonconforming; however, the nonconforming effect does not result in larger errors of numerical solutions as long as a suitable coupling strategy is used.