A fourth order accurate difference-analytical method for solving Laplace’s boundary value problem with singularities

On solving the cracked‐beam problem by block method

“…In Table III, the values of the solution v(x, y) of the cracked-beam problem reported in the literature are given at the three chosen points of Table I. In the last column of Table III, the given results are obtained by a new combined BGM (see [5,21,26,27]), which is the combination of the BM and the finite difference method. The comparison of Table III with the given results in Table I shows the extreme accuracy of the results obtained by BM.…”

“…The comparisons of the results obtained in this paper, with the above-mentioned papers show that the BM gives high accurate results not only around the singular point for the solution and the stress intensity factor, but also everywhere on the closed domain. Therefore, BM can approximate more accurately the other important quantities such as capacitance which is computed by integrating the normal derivative of the solution on the boundary, where the essential boundary condition is given (see [8,18] Other results for the cracked-beam problem are given in [2,[19][20][21]. Some problems of the Laplace equation involving singularities were solved by one-block version of BM in [17,[22][23][24], by four blocks in [25] and by combined block-grid method (BGM) in [5,21].…”

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

“…the given function in (1). We use the matching operator 4 at the points of the set ℎ, constructed in [14]. The value of 4 ( ℎ , ) at the point ∈ ℎ, is expressed linearly in terms of the values of ℎ at the points of the grid constructed on Π ( ) , ( ∈ Π ( ) ) some part of whose boundary located in is the maximum distance away from , and in terms of the boundary values of ( ) , = 0, 1, 2, 3 at a fixed number of points.…”

“…In [2][3][4][5][6][7], a new combined difference-analytical method, called the block-grid method (BGM), is proposed for the solution of the Laplace equation on polygons, when the boundary functions on the sides causing the singular vertices are given as algebraic polynomials of the arc length. In the BGM, the given polygon is covered by a finite number of overlapping sectors around the singular vertices ("singular" parts) and rectangles for the part of the polygon which lies at a positive distance from these vertices ("nonsingular" part).…”

On the Order of Maximum Error of the Finite Difference Solutions of Laplace’s Equation on Rectangles

“…In many versions of the domain decomposition, composite grids and combined methods in solving Laplace's boundary value problems, the obtained system of equations is separated into a fixed number of subsystems, each of which is adequate for the difference equations on a rectangle (see [2,3,5,6,8,10,14]). Therefore, a detailed error analysis becomes important for the classical finite difference or finite element methods for the problems in rectangular domains.…”