The GMRES solver for the interpolating meshless local Petrov–Galerkin method applied to heat conduction

Abstract

Purpose

The work presents a novel implementation of the generalized minimum residual (GMRES) solver in conjunction with the interpolating meshless local Petrov–Galerkin (MLPG) method to solve steady-state heat conduction in 2-D as well as in 3-D domains.

Design/methodology/approach

The restarted version of the GMRES solver (with and without preconditioner) is applied to solve an asymmetric system of equations, arising due to the interpolating MLPG formulation. Its performance is compared with the biconjugate gradient stabilized (BiCGSTAB) solver on the basis of computation time and convergence behaviour. Jacobi and successive over-relaxation (SOR) methods are used as the preconditioners in both the solvers.

Findings

The results show that the GMRES solver outperforms the BiCGSTAB solver in terms of smoothness of convergence behaviour, while performs slightly better than the BiCGSTAB method in terms of Central processing Unit (CPU) time.

Originality/value

MLPG formulation leads to a non-symmetric system of algebraic equations. Iterative methods such as GMRES and BiCGSTAB methods are required for its solution for large-scale problems. This work presents the use of GMRES solver with the MLPG method for the very first time.

Citation

Singh, A.K. and Singh, K.M. (2022), "The GMRES solver for the interpolating meshless local Petrov–Galerkin method applied to heat conduction", Engineering Computations, Vol. 39 No. 2, pp. 493-522. https://doi.org/10.1108/EC-01-2021-0067