Simulating transient phenomena via residual free bubbles A.L.G.A. Coutinho Center for Parallel Computing and Department of Civil Engineering COPPE/Federal University of Rio de Janeiro P.O. Box 68506 Rio de Janeiro, RJ 21945, Brazil L. P. Franca University of Colorado Denver Department of Mathematical and Statistical Sciences P.O.Box 173364, Campus Box 170 Denver, Colorado 80217-3364 F. Valentin National Laboratory for Scientific Computing - LNCC Av. Getúlio Vargas, 333 25651-070 Petrópolis, RJ, Brazil Abstract We derive two stabilized methods for transient equations using static condensation of residual-free bubbles. The methods enhance the stability of the Discontinuous Galerkin method. 1 Introduction Time dependent problems are generally discretized using finite elements in space and finite difference methods in time. This is termed a semidiscrete formulation. An exception to this approach is to fully discretize, i.e., use finite elements both in time and space. In order to make it feasible Discontinuous Galerkin method is used in time. In this paper we start with the Discontinuous Galerkin method in time for a couple of model problems using piecewise linear approximations in time and in space, then we enrich the trial and test functions using residual-free bubbles [1, 2] and we use static condensation to derive stabilized methods. We would like to point out that this elimination is only possible because we add a temporal artificial diffusion that is taken to be zero at the end of the derivation of the methods. Temporal artificial diffusion (TAD) is not a common notion as spatial artificial diffusion (SAD). TAD has been introduced as an elliptic regularization in time [3, 4]. In the next section we use this idea for an initial value problem and in the subsequent section we apply it to a purely advective problem. 2 Initial Value Problem Consider the IVP given by: find u(t) such that u ,t = f (t) in (0,T ] u(0) = u 0 , (1) where f (t) is a piecewise constant function defined in a partition of (0,T ] into subintervals I n =(t n ,t n+1 ),n = 0, 1, ..., N - 1 with t N = T . The solution to (1) is straightforward and it follows from the Fundamental Theorem of Calculus as u(t)= u 0 +  t 0 f (τ ) dτ (2) Herein we are interested in numerical approximations of (1). 1