Continuous Interior Penalty Finite Element Method And Interior Penalty Discontinuous Galerkin Method For Helmholtz Equation With High Wave Number

The numerical simulation of wave scattering problems with high frequencies (or large wave numbers) are significant in applications such as sound waves, electromag-netic waves, shallow water wave field. Designing and analyzing high performance algorithms is the legacy of the well-known open problem in the last century (cf.[84]). Although some significant progresses in the understanding of the behavior of numeri-cal methods for wave scattering problems have been made in the past, this well-known open problem still remain to be unresolved.This dissertation addresses the pre-asymptotic stability and error estimates of the continuous interior penalty finite element methods (CIP-FEMs) including FEM and interior penalty discontinuous Galerkin (IPDG) methods for the Helmholtz equation with large wave number:firstly, we study the linear CIP-FEM for Helmholtz equation in one dimension; secondly, the analysis of hp versions of some CIP-FEM and the FEM for the Helmholtz equation with large wave number in two and three dimensions is continued; finally, the hp-IPDG method proposed in [47] is further discussed in detail.In the first part, the linear version of the CIP-FEM is applied to a one-dimensional model problem. Subtle stability and error estimates are derived. It is proved that the pollution error can be eliminated by selecting the penalty parameter appropriately. Nu-merical results are given to demonstrate the effectiveness of the proposed CIP-FEM.In the second part, we consider hp version of some CIP-FEM for the Helmholtz scattering problem with high wave number in two and three dimensions. By using a "modified duality argument", the first error estimates with pollution term are derived for high order CIP-FEM and FEM. It is shown that the pollution errors of both methods coincide with the existent phase error by the dispersion analysis for the FEM on Carte-sian grids. Moreover, it is proved that the CIP-FEM is absolutely stable (that is, stable for any k, h. and p) by using the so-called "Oswald interpolation"[22,66,73].In the third part, by using the "modified duality argument" in the previous part, the pre-asymptotic stability and error estimates of hp version of the IPDG method from [47] are further improved.