A formalism for designing robust multigrid methods for Helmholtz-like problems

This paper discusses templates for designing efficient and robust multigrid (MG) methods. All MG mechanisms are described by means of matrix functions (block symbols), which capture the properties of the full matrices. We propose a new strategy for deriving the block symbol by means of sparse matrices and of its representation in terms of the usual tensor product. Emphasis is laid on the indefinite Helmholtz equation, which is of great practical interest in real-life applications. We take advantage of previously known patterns (the subblock smoother as preconditioner and the full projection as grid transfer operator) for designing MG methods that work as direct solvers. The corresponding numerical results are presented and the superiority of the combined use of a subblock smoother and a full projection is demonstrated. The condition number and positive definiteness of the subblock smoother especially for Helmholtz-like problems are analyzed, and the corresponding requirements and bounds are given.