MATHEMATICS OF COMPUTATION Volume 72, Number 244, Pages 1689–1709 S 0025-5718(03)01516-3 Article electronically published on May 21, 2003 A NONCONFORMING COMBINATION OF THE FINITE ELEMENT AND VOLUME METHODS WITH AN ANISOTROPIC MESH REFINEMENT FOR A SINGULARLY PERTURBED CONVECTION-DIFFUSION EQUATION SONG WANG AND ZI-CAI LI Abstract. In this paper we formulate and analyze a discretization method for a 2D linear singularly perturbed convection-diffusion problem with a singular perturbation parameter ε. The method is based on a nonconforming combi- nation of the conventional Galerkin piecewise linear triangular finite element method and an exponentially fitted finite volume method, and on a mixture of triangular and rectangular elements. It is shown that the method is stable with respect to a semi-discrete energy norm and the approximation error in the semi-discrete energy norm is bounded by Ch      ln ε ln h     with C independent of the mesh parameter h, the diffusion coefficient ε and the exact solution of the problem. 1. Introduction Many phenomena in engineering, physics and finance are governed by a convec- tion-diffusion equation with a diffusion coefficient 1 >ε> 0 which is much smaller than the average magnitude of the convection coefficient function. These equations are convection-dominated or singularly perturbed, and solutions to them normally have sharp boundary or interior layers so that applications of conventional numer- ical methods to these problems often yield solutions with nonphysical, spurious os- cillations. To overcome this stability problem, many methods have been proposed. These include upwind methods (cf., for example, [5, 11, 12, 4, 2]), streamline dif- fusion methods (cf., for example, [13]) and exponentially fitted methods (cf., for example, [14, 15, 18, 23, 24]). However, no method guarantees, in general, that a numerical solution converges to the exact one uniformly in ε on an unstructured tri- angular partition. Several uniformly convergent schemes (cf., for example, [18, 22]) have been proposed and analyzed based on the Shishkin mesh technique (cf. [16]) and on piecewise uniform, structured partitions. Thus, these schemes can hardly be used for problems with nonrectangular geometries. Work on a least-squares finite element using a Gartland-type mesh is reported in [21]. Recently, Feistauer et al. ([7, 8, 9]) proposed and analyzed some semi-implicit and explicit schemes based on Received by the editor June 7, 2001 and, in revised form, December 28, 2001. 2000 Mathematics Subject Classification. Primary 65N30; Secondary 76M10. c 2003 American Mathematical Society 1689 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use