Some upwinding techniques for finite element approximations of convection-diffusion equations

Abstract

Summary.A uniform framework for the study of upwinding schemes is developed. The standard finite element Galerkin discretization is chosen as the reference discretization, and differences between other discretization schemes and the reference are written as artificial diffusion terms. These artificial diffusion terms are spanned by a four dimensional space of element diffusion matrices. Three basis matrices are symmetric, rank one diffusion operators associated with the edges of the triangle; the fourth basis matrix is skew symmetric and is associated with a rotation by g/2. While finite volume discretizations may be written as upwinded Galerkin methods, the converse does not appear to be true. Our approach is used to examine several upwinding schemes, including the streamline diffusion method, the box method, the Scharfetter-Gummel discretization, and a divergence-free scheme.

A monotone finite element scheme for convection-diffusion equations

“…Standard finite element and/or finite difference methods are in general not suitable for these problems, in the sense that the numerical solution often contains spurious oscillations if the mesh size is not small enough. Many special techniques have been developed, including upwinding finite difference and/or finite volume methods (see [3], and [4]), finite volume methods (see [13]), streamline diffusion finite element methods [17], the Petrov-Galerkin method (see [16]), and (the hybrid) streamline-upwinding-Petrov-Galerkin (SUPG) method (see [11] and [16]). For a detailed description of numerical techniques and analytical tools in investigating convection-diffusion equations we refer to the monographs [23] and [24].…”

Error Estimates of the Finite Element Method with Weighted Basis Functions for a Singularly Perturbed Convection-Diffusion Equation

“…To avoid non-physical numerical solutions, many special finite element techniques have been developed, including upwind finite element [1,4], Petrov-Galerkin finite element [7], streamline diffusion finite element methods [2,8,9], and exponentially fitted finite elements [18,[21][22][23]. However, these methods do not always give accurate results, especially when a diffusion coefficient has the same magnitude as that of mesh size.…”

A nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection-diffusion equation

“…To overcome this stability problem, many methods have been proposed. These include upwind methods (cf., for example, [5,11,12,4,2]), streamline diffusion 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 triangular partition.…”