FEM For Helmholtz Equation With High Wave Number And Perfectly Matched Layer Truncation

The Helmholtz scattering problem with high wave number is truncated by the perfectly matched layer(PML)technique and then discretized by finite element method(FEM)and continuous interior penalty finite element method(CIP-FEM).It is proved that the truncated PML problem satisfies the inf-sup condition with inf-sup constant of order O(k-1).Stability and convergence of the truncated PhIL problem are discussed.In particular,the convergence rate is better than the previous result.The prcasymptotic error estimates in the energy norm of the linear CIP-FEM as well as FEM are proved to be C1kh+C2k3h2 under the mesh condition that k3h2 is sufficiently small.Especially,the estimates of the p-degree(CIP-)FEM arc discussed for the one dimensional Helmholt.z problem when kp+2hp+1 is small enough.Numerical tests arc provided to illustrate the prcasymptotic error estimates and show that the penalty parameter in the CIP-FEM may be tuned to reduce greatly the pollution error.A posteriori error analysis,which gives both the upper bounds and lower bounds of the error of finite element(FE)solution according to the a posteriori error indicator,is derived for the truncated PML problem.It is proved that the residual-type indicator indicates just the error of the elliptic projection but not the pollution error of the FE solution.Xumorical tests of the standard adaptiveFEM are performed and show that the H1-error is proportional to N-1/2 when k is small,where N is the number of degrees,but suffers from the pollution error when k is large.Finally,the nonlinear Helmholtz equation with PML truncation is discussed.The stability and convergence of the nonlinear PML solution are derived with explicit dependence on the wave number under the condition that the parameter of nonlinear medium is sufficiently small.The well-posedness and prcasymptotic error estimates of the linear FEM and its iterative methods are achieved,and some numerical tests for the relevant,problem of nonlinear optics are carried out.