Pre-asymptotic error analysis of hp-interior penalty discontinuous Galerkin methods for the Helmholtz equation with large wave number

In this paper we shall consider to improve the pre-asymptotic stability and error estimates of some h p -interior penalty discontinuous Galerkin ( h p -IPDG) methods for the Helmholtz equation with the first order absorbing boundary condition in two and three dimensions given in Feng and Wu (2011). The proposed h p -IPDG methods are defined using a sesquilinear form which is not only mesh-dependent (or h -dependent) but also degree-dependent (or p -dependent). By using a modified duality argument given in Zhu and Wu (2013), pre-asymptotic error estimates are improved for the proposed h p -IPDG methods under the condition of k h p ? C 0 ( p k ) 1 p + 1 in this paper, where C 0 is some constant independent of k , h , p , and the penalty parameters. It is shown that the pollution error of the method in the broken H 1 -norm is O ( k 2 p + 1 h 2 p ) if p = O ( 1 ) which coincides with existent dispersion analyses for the DG method on Cartesian grids. Numerical tests are provided to verify the theoretical findings and to illustrate great capability of the IPDG method in reducing the pollution effect.

[1]  Jie Shen,et al.  Analysis of a Spectral-Galerkin Approximation to the Helmholtz Equation in Exterior Domains , 2007, SIAM J. Numer. Anal..

[2]  Mark Ainsworth,et al.  Dispersive and Dissipative Properties of Discontinuous Galerkin Finite Element Methods for the Second-Order Wave Equation , 2006, J. Sci. Comput..

[3]  I. Babuska,et al.  Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM☆ , 1995 .

[4]  John C. Strikwerda,et al.  Interior Regularity Estimates for Elliptic Systems of Difference Equations , 1983 .

[5]  I. Babuska,et al.  Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions , 1999 .

[6]  Isaac Harari,et al.  Reducing spurious dispersion, anisotropy and reflection in finite element analysis of time-harmonic acoustics , 1997 .

[7]  Alexandre Ern,et al.  Discrete functional analysis tools for Discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations , 2010, Math. Comput..

[8]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[9]  A. K. Aziz,et al.  On the Numerical Solutions of Helmholtz’s Equation by the Finite Element Method , 1980 .

[10]  Mark Ainsworth,et al.  Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods , 2004 .

[11]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[12]  Jens Markus Melenk,et al.  Wavenumber Explicit Convergence Analysis for Galerkin Discretizations of the Helmholtz Equation , 2011, SIAM J. Numer. Anal..

[13]  Jim Douglas,et al.  APPROXIMATION OF SCALAR WAVES IN THE SPACE-FREQUENCY DOMAIN , 1994 .

[14]  Haijun Wu,et al.  Preasymptotic Error Analysis of Higher Order FEM and CIP-FEM for Helmholtz Equation with High Wave Number , 2014, SIAM J. Numer. Anal..

[15]  Victor M. Calo,et al.  A class of discontinuous Petrov-Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D , 2011, J. Comput. Phys..

[16]  A. Majda,et al.  Radiation boundary conditions for acoustic and elastic wave calculations , 1979 .

[17]  Mark Ainsworth,et al.  Discrete Dispersion Relation for hp-Version Finite Element Approximation at High Wave Number , 2004, SIAM J. Numer. Anal..

[18]  A. H. Schatz,et al.  An observation concerning Ritz-Galerkin methods with indefinite bilinear forms , 1974 .

[19]  I. Babuska,et al.  Nonconforming Elements in the Finite Element Method with Penalty , 1973 .

[20]  Haijun Wu,et al.  hp-Discontinuous Galerkin methods for the Helmholtz equation with large wave number , 2008, Math. Comput..

[21]  Lexing Ying,et al.  Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers , 2010, Multiscale Model. Simul..

[22]  B. Rivière,et al.  Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I , 1999 .

[23]  Leszek Demkowicz,et al.  Wavenumber Explicit Analysis of a DPG Method for the Multidimensional Helmholtz Equation , 2011 .

[24]  Haijun Wu,et al.  Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number , 2009, SIAM J. Numer. Anal..

[25]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[26]  Ivo Babuška,et al.  A Generalized Finite Element Method for solving the Helmholtz equation in two dimensions with minimal pollution , 1995 .

[27]  D. Arnold An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

[28]  R. B. Kellogg,et al.  A scattering problem for the Helmholtz equation , 1979 .

[29]  C. Goldstein,et al.  The finite element method with non-uniform mesh sizes applied to the exterior Helmholtz problem , 1982 .

[30]  Eric T. Chung,et al.  Optimal Discontinuous Galerkin Methods for Wave Propagation , 2006, SIAM J. Numer. Anal..

[31]  Jens Markus Melenk,et al.  Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions , 2010, Math. Comput..

[32]  U. Hetmaniuk Stability estimates for a class of Helmholtz problems , 2007 .

[33]  Alexandre Ern,et al.  Discontinuous Galerkin approximation with discrete variational principle for the nonlinear Laplacian , 2008 .

[34]  L. Thompson A review of finite-element methods for time-harmonic acoustics , 2006 .

[35]  Yulong Xing,et al.  Absolutely stable local discontinuous Galerkin methods for the Helmholtz equation with large wave number , 2010, Math. Comput..

[36]  Harold L. Atkins,et al.  Two-dimensional Wave Analysis of the Discontinuous Galerkin Method with Non-Uniform Grids and Boundary Conditions , 2002 .

[37]  Harold L. Atkins,et al.  Eigensolution analysis of the discontinuous Galerkin method with non-uniform grids , 2001 .

[38]  Gang Bao,et al.  Finite element approximation of time harmonic waves in periodic structures , 1995 .

[39]  Peter Monk,et al.  The numerical solution of the three-dimensional inverse scattering problem for time harmonic acoustic waves , 1987 .

[40]  B. Engquist,et al.  Computational high frequency wave propagation , 2003, Acta Numerica.

[41]  Fernando A. Rochinha,et al.  A discontinuous finite element formulation for Helmholtz equation , 2006 .

[42]  Spencer J. Sherwin,et al.  Dispersion Analysis of the Continuous and Discontinuous Galerkin Formulations , 2000 .

[43]  J. Douglas,et al.  Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods , 1976 .

[44]  O. C. Zienkiewicz,et al.  Achievements and some unsolved problems of the finite element method , 2000 .

[45]  A. H. Schatz,et al.  Interior estimates for Ritz-Galerkin methods , 1974 .

[46]  Yu-Lin Chou Applications of Discrete Functional Analysis to the Finite Difference Method , 1991 .

[47]  Andrew M. Stuart,et al.  Discrete Gevrey regularity attractors and uppers–semicontinuity for a finite difference approximation to the Ginzburg–Landau equation , 1995 .

[48]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[49]  I. Babuska,et al.  Finite Element Solution of the Helmholtz Equation with High Wave Number Part II: The h - p Version of the FEM , 1997 .

[50]  Jens Markus Melenk,et al.  General DG-Methods for Highly Indefinite Helmholtz Problems , 2013, J. Sci. Comput..

[51]  Haijun Wu,et al.  Linear continuous interior penalty finite element method for Helmholtz equation With High Wave Number: One‐Dimensional Analysis , 2012, 1211.1424.

[52]  Xuejun Xu,et al.  A Hybridizable Discontinuous Galerkin Method for the Helmholtz Equation with High Wave Number , 2012, SIAM J. Numer. Anal..

[53]  G. A. Baker Finite element methods for elliptic equations using nonconforming elements , 1977 .

[54]  Stefan A. Sauter,et al.  Is the Pollution Effect of the FEM Avoidable for the Helmholtz Equation Considering High Wave Numbers? , 1997, SIAM Rev..

[55]  Haijun Wu,et al.  Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part I: linear version , 2014 .

[56]  F. Ihlenburg Finite Element Analysis of Acoustic Scattering , 1998 .

[57]  P. Cummings,et al.  SHARP REGULARITY COEFFICIENT ESTIMATES FOR COMPLEX-VALUED ACOUSTIC AND ELASTIC HELMHOLTZ EQUATIONS , 2006 .

[58]  Alexandre Ern,et al.  Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations , 2007, Math. Comput..

[59]  Xiaobing Feng,et al.  Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition , 2007, Math. Comput..

[60]  Thomas J. R. Hughes,et al.  Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains , 1992 .

[61]  B. Rivière,et al.  Part II. Discontinuous Galerkin method applied to a single phase flow in porous media , 2000 .

[62]  C. L. Chang,et al.  A least-squares finite element method for the Helmholtz equation , 1990 .

[63]  Béatrice Rivière,et al.  Discontinuous Galerkin methods for solving elliptic and parabolic equations - theory and implementation , 2008, Frontiers in applied mathematics.

[64]  Xiaobing Feng,et al.  Discontinuous Galerkin finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations , 2013, J. Comput. Appl. Math..

[65]  M. Y. Hussaini,et al.  An Analysis of the Discontinuous Galerkin Method for Wave Propagation Problems , 1999 .

[66]  P. Pinsky,et al.  Complex wavenumber Fourier analysis of the p-version finite element method , 1994 .

[67]  Haijun Wu,et al.  Preasymptotic Error Analysis of CIP-FEM and FEM for Helmholtz Equation with High Wave Number. Part II: hp Version , 2012, SIAM J. Numer. Anal..