GMRES Solver for MLPG Method Applied to Heat Conduction

In the case of large scale simulations involving complex domain, the task of mesh generation is quite a challenging and tedious task both in terms of time and human effort. In recent years, significant research efforts have been invested in the development of mesh-free methods to solve engineering and scientific problems without mesh generation. These methods also overcome the shortcomings of the mesh-based methods such as element distortion in case of problems involving large deformations and crack propagation. Of these methods, the meshless local Petrov Galerkin (MLPG) method is amongst the noticeable ones.  MLPG method is a promising meshfree method for continuum problems in complex domains, especially for large deformation, moving boundary and phase change problems.

However, most of the applications have been limited to small scale problems which are usually solved using direct solvers. Direct solvers such as the Gauss elimination method or LU decomposition method are of the complexity of O(N³), where N is the order of the stiffness matrix. There are large scale problems, which have to be modeled and simulated with high accuracy and results in millions of discretization points. The use of direct solvers is not practical for such large scale problems. Very least number of attempts have been made to solve large scale complex geometry problems using MLPG approach which needs many millions (or even billions) of computing nodes and would require fast iterative solvers. The MLPG method leads to the asymmetric system of algebraic equations. Bi-Conjugate Gradient Stabilised (BiCGSTAB) and Generalized Minimum Residual (GMRES) solvers are the most popular Krylov subspace solvers for an asymmetric linear system of equations. The convergence rate of these methods depends on the preconditioner used. There are basic iterative solvers such as Jacobi and Gauss-Seidel, but these are very slow in convergence. However, these basic methods can still be used as a preconditioner for Krylov subspace solvers.    

In the present work, restarted version of the GMRES method, with and without preconditioner, is applied in connection with the Interpolating MLPG method. Two dimensional and three-dimensional linear steady-state heat conduction problems are taken as model test problems. The error has been calculated based on the exact solution of these problems. Its performance is compared with the preconditioned BiCGSTAB method based on computation time and convergence behavior. Jacobi and successive over-relaxation (k) methods are used as a preconditioner in both the solvers. This work is an attempt to develop a robust and fast MLPG solver.