A fast algorithm for solving the dirichlet problem on hexagonal templet for the poisson equation in a rectangle

Abstract

Using an explicit form of eigenvalues of the Laplacian on a hexagonal molecule, an economical method based on a fast Fourier transform is constructed for solving the Dirichlet problem in a rectangle. Bibliography: 8 titles.Solution of many problems of science and technology involving approximation of partial differential equations calls for the use of hexagonal nets [1]. From a mathematical point of view, the advantage of these nets lies, for instance, in that the fourth-order-error approximation of a difference scheme is attained on a seven-point templet; in the case of a rectangular net, on a nine-point templet [2]. As was shown in [3], the greater closeness of a hexagon, in comparison with a rectangle, to a circle results in better dispersion properties of the difference scheme for the wave equation, which makes use of hexagonal templet for the Laplacian.At the same time, certain difficulties arising in approximation of boundary conditions and the absence of effective algorithms for realization of difference schemes limited both the investigation and wide practical implementation of schemes with hexagonal molecule.In the case of the Dirichlet problem for the Poisson equation in a rhombus, there was constructed in [4] a scale of consistent estimates for the convergence rate of a difference scheme on a hexagonal templet, which are similar to those for a rectangle with the use of a rectangular templet [5]. In [3] the eigenfunctions and eigenvalues of the Laplacian difference operator on a hexagonal net were found explicitly for the Dirichlet problem in a rectangle. Estimates for the maximal and minimal eigenvalues were also obtained. The dispersion of an explicit difference scheme for a two-dimensionM wave equation with a hexagonal templet for the Laplacian was analyzed. Some iteration methods for solving difference problems for equations of the diffusion type on hexagonal nets were considered in [6].In this paper a method for solving the Dirichlet problem in a rectangle is developed, which makes use of an explicit form of the eigenvalues of the Laplacian on a hexagonal templet [3] and an algorithm for the fast Fourier transform [7]. Let us consider the boundary-value problemwhere f2 = {0 < xl < 1, 0 < z2 < v~/2}. To construct a difference scheme, the domain f2 = f2 U F is covered with a triangular netwhere M is a even number. The boundary of the net domain is ~ = ~ f3 F. At interior points such that all the points of a seven-point templet belong to & the Laplacian is approximated by the difference equation 2