HYBRID FDM-STABILIZED LANCZOS-TYPE IN SOLVING PDE PROBLEMS

Abstract

This study investigates the combination of finite difference method (FDM) and the stabilized Lanczos method to solve various partial differential equation (PDE) problems. This combination is wrapped in the algorithms called hybrid FDMRMEIEMLA and hybrid FDM-RLMinRes. FDM is the discretization method which converts the PDEs into algebraic formula, whereas both RMEIEMLA and RLMinRes are known as the stabilized Lanczos methods in solving largescale problems of SLEs. Their hybrids enable us to find the solutions of PDE problems accurately. There are at least three types of PDEs solved in this study, namely Helmholtz, wave, and heat equations. The convergence rate of our methods computed using the residual norms || b - Axk ||. Numerical results showed that our proposed methods performed well in solving the various PDEs with small residual norms.