Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM☆

摘要: Abstract The paper addresses the properties of finite element solutions for Helmholtz equation. h-version method with piecewise linear approximation is applied to a one-dimensional model problem. New results are shown on stability and error estimation discrete model. In all propositions, assumptions made magnitude hk only, where k wavelength h stepwidth FE-mesh. Previous analytical had been assumption that 2 small. For medium high wavenumber, these do not cover meshsizes in practical applications. main estimate reveals H 1 -norm equation polluted when then quasioptimal; i.e., relation FE-error best generally depends wavenumber . It noted pollution term relative same order as phase lead numerical solution. result this analysis, thorough rigorous understanding behavior throughout range convergence gained. Numerical presented show sharpness estimates highlight some phenomena solution behavior. - p -version FEM studied Part II.