SUMMARYWe propose a multigrid method for solving large‐scale sparse linear systems arising from discretizations of partial differential equations, such as those from finite element and generalized finite difference methods. Our proposed method has the following two characteristics. First, we introduce a hybrid geometric+algebraic multigrid method, or HyGA, to leverage the rigor, accuracy, and efficiency of geometric multigrid (GMG) for hierarchical unstructured meshes, with the flexibility of algebraic multigrid (AMG). Second, we introduce efficient smoothers based on the Chebyshev–Jacobi method for both symmetric and asymmetric matrices. We also introduce a unified derivation of restriction and prolongation operators for weighted‐residual formulations over unstructured hierarchical meshes and apply it to both finite element and generalized finite difference methods. We present numerical results of our method for Poisson equations in both 2‐D and 3‐D and compare our method against the classical GMG and AMG as well as Krylov subspace methods. Copyright © 2014 John Wiley & Sons, Ltd.

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In this paper we introduce the local Modified Successive Overrelaxation (MSOR) method and apply Fourier analysis to study its convergence. Parallelism is introduced by decoupling the mesh points with the use of red-black ordering for the 5-point stencil. The optimum set of values for the parameters involved, when the Jacobi iteration operator possesses imaginary eigenvalues, is determined. The performance of the local MSOR method is illustrated by its application to the numerical solution of the convection diffusion equation. It is found that the proposed method is significantly more efficient than local SOR when the absolute value of the smallest eigenvalue of the Jacobi operator is larger than unity. Finally, the parallel implementation of the local MSOR method is discussed and results are presented for distributed memory processors with a mesh topology.