Subgrid modeling for convection-diffusion-reaction in 1 space dimension using a Haar Multiresolution analysis Johan Hoffman 1 , Claes Johnson 2 and Silvia Bertoluzza 3 1 Courant Institute, 251 Mercer Street, New York, NY-10012, USA 2 Department of Computational Mathematics, Chalmers University, SE-41296 G¨ oteborg, Sweden 3 Istituto di Matematica Applicata e Tecnologie Informatiche del C.N.R. v. Ferrata 1, 27100 Pavia, Italy Abstract In this paper we propose and study a subgrid model for linear convection-diffusion- reaction equations with fractal rough coefficients. The subgrid model is based on scale extrapolation of a modeling residual from coarser scales using a computed solution on a finest scale as reference. We show in experiments that a solution with subgrid model on a scale h in most cases corresponds to a solution without subgrid model on a scale less than h/4. We also present error estimates for the modeling error in terms of modeling residuals. Key words: convection-diffusion-reaction, dynamic subgrid modeling, Haar multiresolution analysis, modeling errors PACS: 47.27.Eq, 47.53.+n 1 Introduction A fundamental problem in science and engineering concerns the mathematical modeling of phenomena involving small scales. This problem arises in molec- ular dynamics, turbulent flow and flow in heterogeneous porous media, for example. Basic models for such phenomena, such as the Schr¨odinger equa- tion or the Navier-Stokes equations, may be very accurate models of the real phenomena but may be so computationally intensive, because of the large number of degrees of freedom needed to represent the small scales, that even computers with power way beyond that presently available may be insufficient Preprint submitted to Elsevier Science 20 November 2003