INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 31: 345–358 (1999) FINITE ELEMENTS FOR CFD—HOW DOES THE THEORY COMPARE? A.J. BAKER a, *, D.J. CHAFFIN b,1 , J.S. IANNELLI b,2 AND S. ROY c,3 a Engineering Science Program, MAES Department, 316A Perkins Hall, Uniersity of Tennessee, Knoxille, TN 37996 -2030, USA b The Uniersity of Tennessee, Knoxille, TN 37996 -2030, USA c J.I. Case Corporation, Burr Ridge, IL, USA SUMMARY The quest continues for accurate and efficient computational fluid dynamics (CFD) algorithms for convection-dominated flows. The boundary value ‘optimal’ Galerkin weak statement invariably requires manipulation to handle the disruptive character introduced by the discretized first-order convection term. An incredible variety of methodologies have been derived and examined to address this issue, in particular, seeking achievement of monotone discrete solutions in an efficient implementation. The UT CFD research group has participated in this search, leading to the development of consistent, encompass- ing theoretical statements exhibiting quality performance, including generalized Taylor series (Lax – Wendroff) methods, characteristic flux resolutions, subgrid embedded high-degree Lagrange bases and assembled stencil optimization for finite element weak statement implementations. For appropriate model problems, recent advances have led to accurate monotone methods with linear basis efficiency. This contribution highlights the theoretical developments and presents quantitative documentation of achieved high-quality solutions. Copyright © 1999 John Wiley & Sons, Ltd. KEY WORDS: finite elements; CFD; Galerkin boundary value; accuracy; convergence 1. INTRODUCTION The computational fluid dynamics (CFD) conservation law system for state variable q = q (x j , t ) is L(q ) = q t +  x j ( f j -f j  ) -s =0 on  ×t R n ×R + , 1 j n, (1) where f j =f (u j , q ) and f j  =f ( q /x j ) are the kinetic and dissipative flux vectors respectively, the convection velocity is u j ,  0 is the diffusion coefficient that varies parametrically and s is a source. Appropriate initial and boundary conditions close system (1) for the well-posed statement. * Correspondence to: Engineering Science Program, MAES Department, 316A Perkins Hall, University of Tennessee, Knoxville, TN 37996-2030 USA. E-mail: ajbaker@cfdlab.engr.utk.edu 1 E-mail: chaffin@lasl.gov 2 E-mail: jiannell@utk.edu 3 E-mail: rsubrata@casecorp.com CCC 0271–2091/99/170345 – 14$17.50 Copyright © 1999 John Wiley & Sons, Ltd.