Convergence Analysis of a Discontinuous Galerkin Method for Wave Equations in Second-Order Form

In this paper we study the convergence property of a spatial discontinuous Galerkin method for wave equations. We prove an optimal convergence rate in the energy norm, which improves an existing su...

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

[2]  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..

[3]  Ivo Babuška,et al.  Validation of A-Posteriori Error Estimators by Numerical Approach , 1994 .

[4]  Zhimin Zhang POLYNOMIAL PRESERVING GRADIENT RECOVERY AND A POSTERIORI ESTIMATE FOR BILINEAR ELEMENT ON IRREGULAR QUADRILATERALS , 2004 .

[5]  Miloš Zlámal,et al.  Superconvergence and reduced integration in the finite element method , 1978 .

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

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

[8]  L. Wahlbin Superconvergence in Galerkin Finite Element Methods , 1995 .

[9]  Xu Yang,et al.  Polynomial Preserving Recovery for High Frequency Wave Propagation , 2017, J. Sci. Comput..

[10]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[11]  Haijun Wu,et al.  Enhancing eigenvalue approximation by gradient recovery on adaptive meshes , 2009 .

[12]  Tie Zhang,et al.  The derivative patch interpolation recovery technique and superconvergence for the discontinuous Galerkin method , 2014 .

[13]  Zhimin Zhang,et al.  Gradient Recovery for the Crouzeix–Raviart Element , 2015, J. Sci. Comput..

[14]  Zhimin Zhang,et al.  Polynomial preserving recovery for anisotropic and irregular grids , 2004 .

[15]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

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

[17]  Ronald H. W. Hoppe,et al.  Finite element methods for Maxwell's equations , 2005, Math. Comput..

[18]  Zhimin Zhang,et al.  A New Finite Element Gradient Recovery Method: Superconvergence Property , 2005, SIAM J. Sci. Comput..

[19]  Thomas Hagstrom,et al.  A New Discontinuous Galerkin Formulation for Wave Equations in Second-Order Form , 2015, SIAM J. Numer. Anal..

[20]  R. Bellman The stability of solutions of linear differential equations , 1943 .

[21]  Zhimin Zhang,et al.  Can We Have Superconvergent Gradient Recovery Under Adaptive Meshes? , 2007, SIAM J. Numer. Anal..