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.

中文翻译：

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用于热传导的插值无网格局部 Petrov-Galerkin 方法的 GMRES 求解器

目的

这项工作提出了广义最小残差 (GMRES) 求解器与插值无网格局部 Petrov-Galerkin (MLPG) 方法相结合的新实现，以求解 2-D 和 3-D 域中的稳态热传导。

设计/方法/方法

GMRES 求解器的重新启动版本（带和不带预处理器）用于求解由于插值 MLPG 公式而产生的不对称方程组。根据计算时间和收敛行为，将其性能与双共轭梯度稳定 (BiCGSTAB) 求解器进行比较。Jacobi 和连续过松弛 (SOR) 方法都用作两个求解器中的预处理器。

发现

结果表明，GMRES 求解器在收敛行为的平滑度方面优于 BiCGSTAB 求解器，而在中央处理单元 (CPU) 时间方面的性能略好于 BiCGSTAB 方法。

原创性/价值

MLPG 公式导致代数方程的非对称系统。GMRES和BiCGSTAB方法等迭代方法是解决大规模问题的必要条件。这项工作首次展示了 GMRES 求解器与 MLPG 方法的使用。