Discontinuous Galerkin time stepping with local projection stabilization for transient convection-diffusion-reaction problems N. Ahmed a , G. Matthies b , L. Tobiska a , H. Xie c a Institut f¨ ur Analysis und Numerik, Otto-von-Guericke-Universit¨at Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany b Universit¨at Kassel, Fachbereich 10 Mathematik und Naturwissenschaften, Institut f¨ ur Mathematik, Heinrich-Plett-Straße 40, 34132 Kassel, Germany c LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China Abstract A time-dependent convection-diffusion-reactions problems is discretized in space by a continuous finite element method with local projection stabilization and in time by a discontinuous Galerkin method. We present error estimates for the semidiscrete problem after discretizing in space only and for the fully discrete problem. Numerical tests confirm the theoretical results. Keywords: discontinuous Galerkin, stabilized finite elements, convection-diffusion-reaction equation 1. Introduction The modeling of many technical and physical processes leads to descriptions which contain time-dependent convection-diffusion-reaction equations as sub- problems. Their accurate and efficient solution is often critical for accuracy and efficiency of the whole process. There are several approach for discretizing time-dependent convection-diffusion- reaction problems by finite element methods. Firstly, space-time elements could be used. This results in very large systems of linear equations since all unknowns in the space-time cylinder are coupled. Secondly, semidiscretization as interme- diate steps can be used. Here, we distinguish between the horizontal and vertical methods of lines. The vertical method of lines discretizes first in space and then in time while the horizontal method of lines (or Rothe’s method) applies first a time discretisation which is followed by a discretisation in space. Email addresses: naveed.ahmed@ovgu.de (N. Ahmed), matthies@mathematik.uni-kassel.de (G. Matthies), lutz.tobiska@ovgu.de (L. Tobiska), hhxie@lsec.cc.ac.cn (H. Xie) Preprint submitted to Elsevier October 7, 2010