Staggered discontinuous Galerkin methods for the Helmholtz equation with large wave number

In this paper we investigate staggered discontinuous Galerkin method for the Helmholtz equation with large wave number on general quadrilateral and polygonal meshes. The method is highly flexible by allowing rough grids such as the trapezoidal grids and highly distorted grids, and at the same time, is numerical flux free. Furthermore, it allows hanging nodes, which can be simply treated as additional vertices. By exploiting a modified duality argument, the stability and convergence can be proved under the condition that $\kappa h$ is sufficiently small, where $\kappa$ is the wave number and $h$ is the mesh size. Error estimates for both the scalar and vector variables in $L^2$ norm are established. Several numerical experiments are tested to verify our theoretical results and to present the capability of our method for capturing singular solutions.

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

[2]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[3]  Lina Zhao,et al.  A new staggered DG method for the Brinkman problem robust in the Darcy and Stokes limits , 2019, ArXiv.

[4]  Eric T. Chung,et al.  Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwell's equations on Cartesian grids , 2013, J. Comput. Phys..

[5]  D. Peterseim,et al.  Stable Multiscale Petrov-Galerkin Finite Element Method for High Frequency Acoustic Scattering , 2015, 1503.04948.

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

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

[8]  Eric T. Chung,et al.  A Staggered Discontinuous Galerkin Method for the Stokes System , 2013, SIAM J. Numer. Anal..

[9]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[10]  Lina Zhao,et al.  Guaranteed A Posteriori Error Estimates for a Staggered Discontinuous Galerkin Method , 2018, J. Sci. Comput..

[11]  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 .

[12]  Zhimin Zhang,et al.  A Numerical Analysis of the Weak Galerkin Method for the Helmholtz Equation with High Wave Number , 2017 .

[13]  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..

[14]  Lina Zhao,et al.  A lowest-order staggered DG method for the coupled Stokes–Darcy problem , 2020 .

[15]  Peter Monk,et al.  Error Analysis for a Hybridizable Discontinuous Galerkin Method for the Helmholtz Equation , 2011, J. Sci. Comput..

[16]  Lin Mu,et al.  A new weak Galerkin finite element method for the Helmholtz equation , 2015 .

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

[18]  Jeonghun J. Lee,et al.  Analysis of a Staggered Discontinuous Galerkin Method for Linear Elasticity , 2015, Journal of Scientific Computing.

[19]  F. Brezzi,et al.  Basic principles of Virtual Element Methods , 2013 .

[20]  Lina Zhao,et al.  A Staggered Cell-Centered DG Method for Linear Elasticity on Polygonal Meshes , 2020, SIAM J. Sci. Comput..

[21]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[22]  Olof B. Widlund,et al.  Two-Level Overlapping Schwarz Algorithms for a Staggered Discontinuous Galerkin Method , 2013, SIAM J. Numer. Anal..

[23]  R. B. Kellogg,et al.  A two point boundary value problem with a rapidly oscillating solution , 1988 .

[24]  Lina Zhao,et al.  Fully computable bounds for a staggered discontinuous Galerkin method for the Stokes equations , 2018, Comput. Math. Appl..

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

[26]  Peter Monk,et al.  A least-squares method for the Helmholtz equation , 1999 .

[27]  Lina Zhao,et al.  A Staggered Discontinuous Galerkin Method of Minimal Dimension on Quadrilateral and Polygonal Meshes , 2018, SIAM J. Sci. Comput..

[28]  Eric T. Chung,et al.  Optimal Discontinuous Galerkin Methods for the Acoustic Wave Equation in Higher Dimensions , 2009, SIAM J. Numer. Anal..

[29]  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 .

[30]  Emmanuil H. Georgoulis,et al.  A posteriori error estimates for the virtual element method , 2016, Numerische Mathematik.

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

[32]  Ralf Hiptmair,et al.  Plane Wave Discontinuous Galerkin Methods for the 2D Helmholtz Equation: Analysis of the p-Version , 2011, SIAM J. Numer. Anal..

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

[34]  T. Hughes,et al.  Finite element methods for the Helmholtz equation in an exterior domain: model problems , 1991 .

[35]  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..

[36]  Eric T. Chung,et al.  The Staggered DG Method is the Limit of a Hybridizable DG Method , 2014, SIAM J. Numer. Anal..

[37]  Lina Zhao,et al.  A staggered DG method of minimal dimension for the Stokes equations on general meshes , 2019, Computer Methods in Applied Mechanics and Engineering.

[38]  Eric T. Chung,et al.  Analysis of an SDG Method for the Incompressible Navier-Stokes Equations , 2017, SIAM J. Numer. Anal..

[39]  M. Chipot.  Finite Element Methods for Elliptic Problems , 2000 .

[40]  Eric T. Chung,et al.  Mortar formulation for a class of staggered discontinuous Galerkin methods , 2016, Comput. Math. Appl..

[41]  I. Babuska,et al.  GENERALIZED FINITE ELEMENT METHODS — MAIN IDEAS, RESULTS AND PERSPECTIVE , 2004 .

[42]  Dongwoo Sheen,et al.  FREQUENCY DOMAIN TREATMENT OF ONE-DIMENSIONAL SCALAR WAVES , 1993 .

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

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

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

[46]  Lina Zhao,et al.  A priori and a posteriori error analysis of a staggered discontinuous Galerkin method for convection dominant diffusion equations , 2019, J. Comput. Appl. Math..

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

[48]  Yu Du,et al.  Pre-asymptotic error analysis of hp-interior penalty discontinuous Galerkin methods for the Helmholtz equation with large wave number , 2015, Comput. Math. Appl..

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

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

[51]  Weifeng Qiu,et al.  A first order system least squares method for the Helmholtz equation , 2014, J. Comput. Appl. Math..

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