My summer visit to the Center for Applied Scientific Computing (CASC) at Lawrence Livermore National Laboratory (LLNL) was from June 23 to August 13, 1999.
The projects on which I worked were mostly connected with methods for approximate solution of PDEs using domain decomposition on non-matching grids. I worked with Dr. Raytcho Lazarov and Dr. Panayot Vassilevski on two articles, ``Interior Penalty Discontinuous Approximations of Elliptic Problems'' and ``Discontinuous approximations of Advection-Reaction Equations''. The first article studies a penalty discretization of second order elliptic boundary value problems which is first order accurate, if one uses piecewise linear elements. The second one proposes a conservative (finite volume) approximation of advection-reaction problems. I finished the implementation (in C++) for both methods and provided extensive numerical results. I worked on apposteriori error estimators and adaptive refinement and applied some local refinement techniques for the penalty formulation for diffusion problems. Now in progress is work on mortar approximation for convection-dominated diffusion problems (with upwinding), and derivation of local error estimators for purely convection-reaction problems.

August 17, 1999.