Numerical Studies of Galerkin-type Time-discretizations Applied to Transient Convection-diffusion-reaction Equations

We deal with the numerical solution of time-dependent convection-diffusion-reaction equations. We combine the local projection stabilization method for the space discretization with two different time discretization schemes: the continuous Galerkin-Petrov (cGP) method and the discontinuous Galerkin (dG) method of polynomial of degree k. We establish the optimal error estimates and present numerical results which shows that the cGP(k) and dG(k)methods are accurate of order k+ 1, respectively, in the whole time interval. Moreover, the cGP(k)-method is superconvergent of order 2k and dG(k)-method is of order 2k+ 1 at the discrete time points. Furthermore, the dependence of the results on the choice of the stabilization parameter are discussed and compared. Keywords—convection-diffusion-reaction equations, stabilized finite elements, discontinuous Galerkin, continuous Galerkin-Petrov