Finite element methods for second order differential equations with significant first derivatives

Abstract

SUMMARYGalerkin finite element methods based on symmetric pyramid basis functions give poor accuracy when applied to second order elliptic equations with large coefficients of the first order terms. This is particularly so when the mesh size is such that oscillations are present in the numerical solution. In the present note asymmetric linear and quadratic basis functions are introduced and shown to overcome this difficulty in an appropriate two point boundary value problem. In particular symmetric quadratic basis functions are oscillation free and highly accurate for a working range of mesh sizes. INTRODUCI'IONThe numerical solution of second order elliptic partial differential equations is notoriously difficult if first derivatives with sizeable coefficients are present. A good example of such an equation occurs in steady incompressible viscous fluid dynamics where the vorticity transport equation for a two-dimensional problem is where the vorticity w = -[(d2t+b/ax')+(a2t+b/ay2)], is the stream function, u and u are the velocity components, and v is the coefficient of kinematic viscosity. The coefficients of the first order terms in (1) are equivalent to the Reynolds Number and so are large in the majority of realistic problems.If finite difference methods are used to solve problems of this type, central difference approximations to the first derivatives are to be avoided since they give rise to oscillations in the computed solution at reasonable grid sizes. Backward differences based on directions dictated by the coefficients lead to oscillation free solutions but also to a loss in accuracy [Runchal,' Spalding2]. Formulas between central and backward differences are proposed by Barrett.3 Finite element methods with symmetric pyramid basis functions reproduce central difference formulas with their inherent oscillatory properties, although recently Blackburn4 and Miller' have modified the standard linear basis functions to cope with conduction-convection and singular perturbation problems respectively. A recent review of the situation can be found in Zienkiewicz.6

An hp finite element method for convection-diffusion problems in one dimension

“…Two popular approaches to overcoming stability problems are stabilized methods, such as the streamline diffusion method (see, e.g., Johnson & Nävert (1981)) and the related SUPG (see, e.g., Hughes & Brooks (1979) on the one hand, and PG schemes that are based on upwinded test functions on the other hand (see, e.g., Christie et al (1978), Hemker (1977)). The PG idea, originally formulated as an h version, is generalized here to a p and hp setting by introducing appropriate upwinded test functions; in particular, the upwinded test functions of the low-order h version schemes of Christie et al (1978) and Hemker (1977) are special cases of our more general approach.…”

“…The PG idea, originally formulated as an h version, is generalized here to a p and hp setting by introducing appropriate upwinded test functions; in particular, the upwinded test functions of the low-order h version schemes of Christie et al (1978) and Hemker (1977) are special cases of our more general approach. The idea of stabilization can also be pursued in an hp context, and we refer to Melenk & Schwab (1998) for the analysis of the hp streamline diffusion method as a typical representative of stabilized methods.…”

“…The crucial ingredient in the work of Szymczak (1982) and Szymczak & Babu0 ska (1984) (similarly to Christie et al (1978), Hemker (1977)) was the construction of suitable, upwinded test functions by asymptotic analysis of the elemental adjoint problem. The generalization of this asymptotic analysis (and the techniques used by Christie et al (1978), Hemker (1977)) to high-order elements and higher dimensions is not straightforward.…”

“…The perturbation analysis of Section 4.1 shows that fairly weak accuracy requirements on the test functions ψ i j suf"ce to ensure stability of the FEM. Especially for low p, rather`crude' approximations to the L-splines ψ i j are suf"cient; this is the reason why techniques such as the α-quadratic upwinding of Christie et al (1978) (see also Morton (1995) for an up-to-date account on these methods) and the use of Hemker test functions in De Groen (1978), De Groen & Hemker (1979) obtained by freezing coef"cients lead to stable FEM. All these methods are in fact covered by our perturbation analysis.…”

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.…”