Solution of elliptic PDEs by fast Poisson solvers using a local relaxation factor
This paper is introducing new semidirect procedures for solving nonseparable elliptic partial differential equations (PDE). The fast direct solvers (FDS) based on Fourier analysis and cyclic reduction may be used only for separable elliptic PDE. In the case of nonseparable equations semidirect procedures are used, that is iterative procedures driven by a FDS. The new such methods introduced in this paper are based on a one-step iterative procedure: a modified D'Yakonov-Gunn procedure [\textit{E. G. D'Yakonov}, Dokl. Akad. Nauk SSSR 138, 522-525 (1961; Zbl 0102.116)] or a two-step one and a relaxation factor which is gridpoint dependent. For a large class of problems the two-step vs. one-step convergence ratio varies between 2 and 4. Moreover these procedures possess two important properties which the traditional iterative methods lack: the convergence rate is not sensitive to grid cell size and aspect ratio; the convergence rate can be easily estimated using the coefficients of the PDE being solved. PDE with constant and with variable coefficients are investigated. Numerical tests are given for bidimensional problems. A comparison with SOR methods is also made. A three-dimensional problem for incompressible flows in a 180-degree turning channel is also solved numerically.
scientific article; zbMATH DE number 4011845
scientific article