The Analysis of Iterative Elliptic PDE Solvers Based on The Cubic Hermite Collocation Discretization

Abstract. Collocation methods based on bicubic Hermite piecewise polynomials have been proven effective t.echniques for solving second-order linear elliptic PDEs with mixed boundary conditions. The corresponding linear system is in general non-symmetric and non-diagonally dominant. Iterative methods for their solution arc not known and they aTC currently solved using Gauss elimination with scaling and partial pivoting. Point iterat.ive methods do not convcrge even for the collocation equations obtained from model PDE problems. The del/elopment of efficient iterative solvers for these equations is necessary for three-dimensional problems and their parallel solution, since direct solvers tend to be space bound and their parallelization is difficult. In this thesis, we develop block iterative methods for the collocation equations of elliptic PDEs defined on a rectangle and subject to uncoupled mixed boundary conditions. For model problems of this type, we derive analytic expressions for the eigenvalues of the block Jacobi iteration matrix: and determine the optimal parameter for the block SOR method. For the case of general domains, the iterative solution of tile collocation equations is still an open problem. We address this open problem b

A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

In this paper, the fractional order of rational Bessel functions collocation method (FRBC) to solve Thomas-Fermi equation which is defined in the semi-infinite domain and has singularity at $x = 0$ and its boundary condition occurs at infinity, have been introduced. We solve the problem on semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration solves it by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results in other to show that the new method is accurate, efficient and applicable