A hybrid geometric + algebraic multigrid method with semi-iterative smoothers

We 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.