Preasymptotic Error Estimates Of The Finite Element Method For Helmholtz Equation With High Wave Number And The Pure Source Transfer Domain Decomposition Method

This dissertation addresses the preasymptotic error estimates of the continuous interior penalty finite element method (CIP-FEM) (including the standard finite element method) and the interior penalty discontinuous Galerkin method (IPDG) for Helmholtz equation with high wave number and the pure source transfer domain decomposition method (STDDM).We first develop some discrete Sobolev theories on FE spaces. Then we use the elliptic projection Phu of u to decompose the error of the FE solution uh as u-uh= u-Phu+Phu-uh, bound the L2-norm of Phu-uh by its high order discrete Sobolev norms in the duality argument step (instead of its H1-norm as the standard Schatz argument) and prove the preasymptotic error estimates and the stability under the mesh condition that k(hk)2p≤C0.Our numerical tests show CIP-FEM with some special penalty parameters may reduce the pollution errors efficiently.Then we use this kind of arguments to estimate the interior penalty discontinuous Galerkin method and prove the preasymptotic error estimates under the mesh condition that k(hk)2p≤C0. Numerical experiments are provided to verify the theoretical results.To solve the linear system of the finite element method, we give the pure source transfer domain decomposition method proposed recently (pSTDDM). Different from the standard source transfer domain decomposition method, first source transfer and then wave expansion, all of the two steps in our method are the source transfer. Then our method does not need to solve the PML problem in the half-space, make the two steps run in parallel and reduce the errors and computational complexity. Besides, every local PML problem in pSTDDM can be solved by our pSTDDM by recursion, then we can further increase the number of the parallel computation and reduce the linear system of every local problem.At last we consider the dispersion analysis of the continuous interior penalty finite element method, get the phase error estimates between the wave number k of the con-tinuous problem and some discrete wave number kh for different real penalty parame-ters and obtain the "optimal" penalty parameters which are those used in the numerical tests.