Cascadic multigrid methods combined with sixth order compact scheme for poisson equation

Abstract

Based on the extrapolation theory and a sixth order compact difference scheme, new extrapolation interpolation operator and extrapolation cascadic multigrid methods for two dimensional Poisson equation are presented. The new extrapolation interpolation operator is used to provide a better initial value on refined grid. The convergence of the new methods are given. Numerical experiments are shown to illustrate that the new methods have higher accuracy and efficiency.

A Cascadic Multigrid Algorithm on the Shishkin Mesh for a Singularly Perturbed Elliptic Problem with regular layers

“…To construct a better initial guess on refined mesh Ω N we use the same idea of articles [35,36] and the proposed formulas in [29]:…”

“…The two-grid method with the application Richardson extrapolation to increase the ε-uniform accuracy of the difference scheme on the Shishkin mesh is investigated in [25][26][27][28][29][30]33]. In [29,30,33] the multigrid algorithm of the same structure based on three meshes is considered and to reduce the number of iteration we apply the idea of the extrapolation of numerical solutions on coarse meshes [35,36].…”

Sixth order compact multi-phase block-AGE iteration methods for computing 2D Helmholtz equation

“…The Helmholtz equation is pivotal in describing various significant physical phenomena, encompassing the determination of potentials in time-harmonic acoustic and electromagnetic fields, the analysis of acoustic wave scattering, the reduction of noise in silencing systems, the modeling of water wave propagation, the study of membrane vibrations, and the assessment of radar scattering [1] , [2] , [3] , [4] , [5] , [6] . Numerous research endeavors have been directed towards achieving a more efficient and precise numerical solution for the Helmholtz and Poisson equations [7] , [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21] , [22] , [23] , [24] .…”

Improved finite difference method with a compact correction term for solving Poisson’s equations

“…In fact, an improved efficient method for solving partial differential equations can be developed by combining the advantages of two different numerical methods [24][25][26]. Li and Li [27] studied the multigrid method combined with a fourth order compact scheme for the 2D Poisson's equation. The results showed that the new method was of higher accuracy and less computational time.…”