Two-level Algebraic Multigrid for the Helmholtz Problem

An algebraic multigrid method with two levels is applied to the solution of the Helmholtz equation in a first order least squares formulation, discretized by Q1 finite elements with reduced integration. Smoothed plane waves in a fixed set of directions are used as coarse level basis functions. The method is used to investigate numerically the sensitivity of the scattering problem to a change of the shape of the scatterer. Multigrid methods for the solution of the Helmholtz equation of scattering are known in the literature. A common disadvantage of multigrid methods is that the coarsest level must be fine enough to capture the wave character of the problem, or the iterations diverge [6, 7, 10]. One way to overcome this limitation is to use coarse basis functions derived from plane waves [8]. However, manipulating such functions becomes expensive since the cost does not decrease with the number of variables on the coarse levels. It should be noted that functions derived from plane waves can also be used as basis functions for the discretization of the Helmholtz equation itself; such methods are known under the names of the Microlocal Discretization [5], Partition of Unity Finite Element Method [1], or the Finite Element Ray Method [11]. We propose a two-level method with coarse space basis functions defined as a plane waves within an aggregate of nodes, zero outside the aggregate, and then smoothed by a Chebyshev type iteration using the original fine level matrix. This results in a method with good computational complexity and scalability. This method falls under the abstract framework of black-box two-level iterative methods based on the concept of smoothed aggregations [16]. The objective of the smoothing of the coarse basis functions is to reduce their energy [9, 15, 16]. For a related theoretical analysis of such two-level methods with high order polynomial