CAM Seminar—“Nearly normal” preconditioning for the high frequency Helmholtz equation in heterogeneous media
报告人：Shihua Gong (University of Bath)
时间：2019-12-19 15:00-16:00
地点：Room 1418, Sciences Building No. 1
Abstract: The Helmholtz equation with large wave number $k$ is difficult to solve numerically mainly because of the highly oscillatory nature of its solutions (on a scale of $\frac{1}{k}$) and the indefiniteness of its standard variational formulation (with smaller discrete Inf-Sup constant as $k$ increases). High order finite element methods can provide efficient approximation to the oscillatory solutions. The indefiniteness of the arising linear system may be much more severe while taking account the waves in heterogeneous media since some rays may be trapped in local regions and the energy in these regions is significantly large. We consider the OBDD-H (or called as SORAS) preconditioners to solve the linear system arising from the high order discretizations of the heterogeneous Helmholtz equation. The action of these preconditioners requires solutions of independent subproblems with absorbing boundary conditions and (possibly) absorption added in the domain. Then the local solutions are glued together using prolongation operators defined by the composition of a nodal interpolation and a partition of unity. Supporting theory for these preconditioners in the case of homogeneous Helmholtz problems is given by I. G. Graham E. A. Spence and J. Zou in [Preprint arXiv:1806.03731.] In this talk we extend this theory to heterogeneous cases and figure out the effect of the underlying finite element discretizations to the preconditioners. We will explain the importance of “nearly normal” preconditioning for solving the non-self-adjoint problems. We also investigate numerically the role of the `non-trapping' criterion in the performance of the preconditioners. This is a joint work with Ivan G. Graham and Euan A. Spence.
