Stabilized Galerkin-characteristics finite element schemes for flow problems

In this paper stabilized Galerkin-characteristics finite element schemes for the Oseen and the Navier-Stokes equations are analyzed to be stable and convergent. The former is unconditionally stable, and the latter is conditionally stable as a result of the nonlinearity but the condition is revealed to be non-restrictive essentially in real computation. The error estimates of both of the schemes are optimal. By virtue of the property of the characteristics method the schemes are robust for high Reynolds number problems. The schemes employ an inexpensive P1/P1 element which yields a small number of degrees of freedom. Since matrices derived from the schemes are symmetric, powerful linear solvers for symmetric systems of linear equations can be used. The schemes are, therefore, efficient especially for three-dimensional problems. Numerical results from the scheme for the Navier-Stokes equations are presented to show its effectiveness.