Discontinuous Galerkin finite element methods for second order hyperbolic problems

In this paper, we prove a priori and a posteriori error estimates for a finite element method for linear second order hyperbolic problems (linear wave equations) based on using spacetime finite element discretizations (for displacements and displacement velocities) with (bilinear) basis functions which are continuous in space and discontinuous in time. We refer to methods of this form as discontinuous Galerkin methods, or DG methods. The work in this paper generalizes to second order hyperbolic problems our earlier work on DG methods of the same general form (space-time finite element discretizations with the basis functions continuous in space and discontinuous in time) for first order hyperbolic problems and parabolic problems, see [1-4, 8-12] and references therein. DG methods were first applied to second order hyperbolic problems by Hulbert and Hughes [5] and Hulbert [6, 7] with approximations of the displacement u only, or both displacement u and velocity ~. Usually, DG methods are implemented on space-time finite element meshes organized into space-time 'slabs' S n = n x ln, where n is the underlying spacial domain and 1~ =(t~, t~+l) are time intervals corresponding to a sequence {t~} of increasing discrete time levels in. With the corresponding finite element basis functions discontinuous across the discrete time levels t,, the DG method generates an implicit time-stepping method where the approximate solution is sought separately on one slab S~ after the other with n (that is t~) increasing. Introducing additional dependent variables (stresses), second order hyperbolic equations may be written as first order systems to which our previous studies apply. Thus the aim of the present work, as well as that of [5-7], is to keep a displacement formulation, avoiding additional stress variables, which gives a natural extension to dynamic problems of the standard displacement