Discontinuous Time-Space Galerkin Finite Element Discretization Technique for the Analysis of Two-Dimensional Heat Conduction

Abstract

The time-discontinuous Galerkin method based upon using a finite element formulation in time is derived. This method is to use the weighted residuals to treat space and time in a uniform manner and thus integrating both the spatial and temporal variations of the unknown quantities simultaneously. The approximations are continuous with respect to the space variables for each fixed time, but they admit discontinuities with respect to the time variable at each time step. Interpolation functions and weighting functions are taken to be discontinuous across inter-element boundary. This method generates a complete space-time finite element discretization which eliminates the need for any additional ordinary differential equation solver to resolve the temporal behavior of the problem. No significant instability problems and much more rapid convergence to the analytical solution were experienced in this approach than the semidiscretization method.