Least-squares finite element method for the advection–diffusion equation _ Idris Dag ˘ a, * , Dursun Irk b , Mustafa Tombul c a Department of Computer Engineering, Osmangazi University, 26480 Eskis ßehir, Turkey b Department of Mathematics, Osmangazi University, 26480 Eskis ßehir, Turkey c Department of Civil Engineering, Anadolu University, 26470 Eskis ßehir, Turkey Abstract The space–time least-squares finite element methods are constructed for the advec- tion–diffusion equation by using both linear shape functions and quadratic B-spline shape functions. Two test problems are studied to demonstrate the accuracy of the pres- ent methods. Results of the two schemes have been compared. Ó 2005 Elsevier Inc. All rights reserved. 1. Introduction Advection–diffusion equation describes many quantities such as mass, heat, energy, velocity, vorticity, etc. The solutions of this equation model some of the phenomena such as the heat transfer in a draining film, water transfer in soils, spread of pollutants in rivers and streams, contaminant dispersion in shallow lakes, flow in porous media, dispersion of dissolved salts in groundwater, ther- mal pollution in river systems, etc. The slow progress has been made towards 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.04.054 * Corresponding author. E-mail address: idag@ogu.edu.tr ( _ I. Dag ˘). Applied Mathematics and Computation 173 (2006) 554–565 www.elsevier.com/locate/amc