INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2008; 57:1321–1348 Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld.1842 A stabilized mixed finite element method for the first-order form of advection–diffusion equation Arif Masud ∗, †, ‡ and JaeHyuk Kwack § Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 3110 Newmark Civil Engineering Laboratory, MC-250, Urbana, IL 61801-2352, U.S.A. SUMMARY This paper presents a stabilized mixed finite element method for the first-order form of advection–diffusion equation. The new method is based on an additive split of the flux-field into coarse- and fine-scale components that systematically lead to coarse and fine-scale variational formulations. Solution of the fine-scale variational problem is mathematically embedded in the coarse-scale problem and this yields the resulting method. A key feature of the method is that the characteristic length scale of the mesh does not appear explicitly in the definition of the stability parameter that emerges via the solution of the fine-scale problem. The new method yields a family of equal- and unequal-order elements that show stable response on structured and unstructured meshes for a variety of benchmark problems. Copyright 2008 John Wiley & Sons, Ltd. Received 5 October 2007; Revised 5 April 2008; Accepted 10 April 2008 KEY WORDS: stabilized methods; multiscale methods; continuous fields of arbitrary order; advection– diffusion equation; equal- and unequal-order elements 1. INTRODUCTION Advection–diffusion phenomena appear in many problems in physical sciences and engineering, and therefore an accurate modeling of this phenomenon has been a focus of research in the field of fluid mechanics. Advection-dominated diffusion processes are typically modeled via a scalar- valued advection diffusion equation that also serves as a vehicle to study the more advanced flow models, namely, the Navier–Stokes equations. For the advection-dominated case, this equation ∗ Correspondence to: Arif Masud, Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 3110 Newmark Civil Engineering Laboratory, MC-250, Urbana, IL 61801-2352, U.S.A. † E-mail: amasud@uiuc.edu ‡ Associate Professor. § Graduate Research Assistant. Contract/grant sponsor: National Academy of Sciences; contract/grant number: NAS 7251-05-005 Copyright 2008 John Wiley & Sons, Ltd.