Combined DG-CG Time Stepping for Wave Equations

The continuous and discontinuous Galerkin time stepping methodologies are combined to develop approximations of second order time derivatives of arbitrary order. This eliminates the doubling of the number of variables that results when a second order problem is written as a first order system. Stability, convergence, and accuracy, of these schemes is established in the context of the wave equation. It is shown that natural interpolation of nonhomogeneous boundary data can degrade accuracy, and that this problem can be circumvented using interpolants matched with the time stepping scheme.

[1]  Donald A. French,et al.  Continuous finite element methods which preserve energy properties for nonlinear problems , 1990 .

[2]  Thomas J. R. Hughes,et al.  Space-time finite element methods for second-order hyperbolic equations , 1990 .

[3]  R. Showalter.  Hilbert Space Methods for Partial Differential Equations , 1979, Electronic Journal of Differential Equations.

[4]  Charalambos Makridakis,et al.  Convergence of a continuous Galerkin method with mesh modification for nonlinear wave equations , 2004, Math. Comput..

[5]  T. Dupont.  $L^2 $-Estimates for Galerkin Methods for Second Order Hyperbolic Equations , 1973 .

[6]  Donald A. French,et al.  A space-time finite element method for the wave equation* , 1993 .

[7]  Slimane Adjerid,et al.  A discontinuous Galerkin method for the wave equation , 2011 .

[8]  Claes Johnson,et al.  Discontinuous Galerkin finite element methods for second order hyperbolic problems , 1993 .

[9]  J. Rauch.  ON CONVERGENCE OF THE FINITE ELEMENT METHOD FOR THE WAVE EQUATION , 1985 .

[10]  G. A. Baker.  Error Estimates for Finite Element Methods for Second Order Hyperbolic Equations , 1976 .

[11]  T. Hughes,et al.  Space-time finite element methods for elastodynamics: formulations and error estimates , 1988 .

[12]  Peter Monk,et al.  Continuous finite elements in space and time for the heat equation , 1989 .

[13]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[14]  I. Lasiecka,et al.  Negative norm estimates for fully discrete finite element approximations to the wave equation with nonhomogeneous L 2 Dirichlet boundary data , 1995 .

[15]  Slimane Adjerid,et al.  A discontinuous Galerkin method for higher-order ordinary differential equations , 2007 .

[16]  Donald A. French,et al.  A continuous space-time finite element method for the wave equation , 1996, Math. Comput..