It is shown how to discretize the pressure gradient in the Navier{ Stokes equations on an arbitrary non-uniform curvilinear staggered grid, such that the error is zero for constant pressure gradient. A theoretical foundation is given for a discretization proposed in 2] on empirical grounds. A quite similar approach is used to discretize a diffusion equation with strongly discontinuous di usion coe cient, such that the error is zero on a general grid for constant ux.

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We discuss a Petrov-Galerkin mixed finite element formulation of the semiconductor continuity equations on a rectangular domain. We give error estimates for equations that are in principle degenerate in the singularly perturbed case. We give arguments that indicate that the method is also effective in the singularly perturbed case. We develop a discretization that gives a higher-order accurate solution for use in an a posteriori error estimator. 0 1995 John Wiley & Sons, Inc.

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