COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng 2009; 25:787–809 Published online 22 July 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cnm.1156 A variational multiscale model for the advection–diffusion–reaction equation Guillaume Houzeaux ∗, † , Beatriz Eguzkitza and Mariano V´ azquez Barcelona Supercomputing Center (BSC-CNS), Edificio C6-E201, Campus Nord UPC, Jordi Girona 1-3, 08034 Barcelona, Spain SUMMARY The variational multiscale (VMS) method sets a general framework for stabilization methods. By splitting the exact solution into coarse (grid) and fine (subgrid) scales, one can obtain a system of two equations for these unknowns. The grid scale equation is solved using the Galerkin method and contains an additional term involving the subgrid scale. At this stage, several options are usually considered to deal with the subgrid scale equation: this includes the choice of the space where the subgrid scale would be defined as well as the simplifications leading to compute the subgrid scale analytically or numerically. The present study proposes to develop a two-scale variational method for the advection–diffusion–reaction equation. On the one hand, a family of weak forms are obtained by integrating by parts a fraction of the advection term. On the other hand, the solution of the subgrid scale equation is found using the following. First, a two-scale variational method is applied to the one-dimensional problem. Then, a series of approximations are assumed to solve the subgrid space equation analytically. This allows to devise expressions for the ‘stabilization parameter’ , in the context of VMS (two-scale) method. The proposed method is equivalent to the traditional Green’s method used in the literature to solve residual-free bubbles, although it offers another point of view, as the strong form of the subgrid scale equation is solved explicitly. In addition, the authors apply the methodology to high-order elements, namely quadratic and cubic elements. The proposed model consists in assuming that the subgrid scale vanishes also on interior nodes of the element and applying the strategy used for linear element in the segment between these interior nodes. The proposed scheme is compared with existing ones through the solution of a one-dimensional numerical example for linear, quadratic and cubic elements. In addition, the mesh convergence is checked for high-order elements through the solution of an exact solution in two dimensions. Copyright 2008 John Wiley & Sons, Ltd. Received 19 December 2007; Revised 3 April 2008; Accepted 26 May 2008 KEY WORDS: variational multiscale; stabilized finite element; advection–diffusion–reaction equation; high-order elements ∗ Correspondence to: Guillaume Houzeaux, Barcelona Supercomputing Center (BSC-CNS), Edificio C6-E201, Campus Nord UPC, Jordi Girona 1-3, 08034 Barcelona, Spain. † E-mail: guillaume.houzeaux@bsc.es Contract/grant sponsor: Spanish Project OPTIDIS; contract/grant number: ENE2005-05274 Contract/grant sponsor: Spanish Ministerio de Educaci´ on y Ciencia Copyright 2008 John Wiley & Sons, Ltd.