IPCDGM and multiscale IPDPGM for the Helmholtz problem with large wave number

Abstract

In this paper, based on the two nested spaces of piecewise linear polynomials (on fine and coarse meshes with sizes h and H respectively), which are continuous in macro elements but discontinuous across interior edges/faces of them, an interior penalty continuous–discontinuous Galerkin method (IPCDGM) and a multiscale interior penalty discontinuous Petrov–Galerkin method (MsIPDPGM) are proposed to solve the Helmholtz problem with large wave number k and homogeneous Robin boundary condition. This paper devotes to analyzing the preasymptotic error of the two methods separately. In order to reduce the pollution errors efficiently, the two methods not only include the penalty terms on jumps of function on coarse mesh edges/faces but also add the penalty terms on jumps of first order normal derivatives on fine mesh edges/faces. The error of the IPCDG solution in the broken H1-norm is bounded by O(kh+k3h2). By splitting into coarse and fine scales and basing on the IPCDG variational formulation, the MsIPDPGM’s trial and test spaces are constructed with the macro corrected bases. Due to the exponential decay of the correctors, the corrected bases are obtained by solving the local problems on localized subdomains of size LH (L being the oversampling parameter). The preasymptotic error analysis of MsIPDPGM shows that, if kH and logk∕L fall below some constants and if the fine mesh size h is sufficient small, the errors of numerical solutions in the broken H1-norm can be dominated by O(H3) without pollution effect. Numerical tests are provided to verify the theoretical findings and advantages of the two methods.