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

The present study was performed to determine whether nitric oxide overproduction is associated with deterioration in peripheral nerve function in type 1 diabetes. We measured peripheral nerve function and biochemical indicators of nitrosative stress annually for 3 years in 37 patients with type 1 diabetes. Plasma nitrite and nitrate (collectively NO x ) were 34.0 ؎ 4.9 mol/l in the control subjects and 52.4 ؎ 5.1, 50.0 ؎ 5.1, and 49.0 ؎ 5.2 in the diabetic patients at the first, second, and third evaluations, respectively (P < 0.01). Nitrotyrosine (NTY) was 13.3 ؎ 2.0 mol/l in the control subjects and 26.8 ؎ 4.4, 26.1 ؎ 4.3, and 32.7 ؎ 4.3 in the diabetic patients (P < 0.01). Uric acid was suppressed by 20% in the diabetic patients (P < 0.001). Composite motor nerve conduction velocity for the median, ulnar, and peroneal nerves was decreased in patients with high versus low NTY (mean Z score ؊0.522 ؎ 0.25 versus 0.273 ؎ 0.22; P < 0.025). Patients with high NO x had decreased sweating, and those with suppressed uric acid had decreased autonomic function. In conclusion, nitrosative stress in early diabetes is associated with suppressed uric acid and deterioration in peripheral nerve function.

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