On the Stability of Residual-free Bubbles for Convection-diiusion Problems and Their Approximation by a Two-level Nite Element Method

We consider the Galerkin nite element method for partial diiferential equations in two dimensions, where the nite-dimensional space used consists of piecewise (isoparametric) polynomials enriched with bubble functions. Writing L for the diierential operator, we show that for elliptic convection-diiusion problems, the component of the bubble enrichment that stabilizes the method is equivalent to a Petrov-Galerkin method with an L-spline (exponentially tted) trial space and piecewise polynomial test space; the remaining component of the bubble innuences the accuracy of the method. A stability inequality recently obtained by Brezzi, Franca and Russo for a limiting case of bubbles applied to convection-diiusion problems is shown to be slightly weaker than the standard stability inequality that is obtained for the SDFEM/SUPG method, thereby demonstrating that the bubble approach is in general slightly less stable than the streamline diiusion method. When the trial functions are piecewise linear, we show that residual-free bubbles are as stable as SDFEM/SUPG, and we extend this stability inequality to include positive mesh-Peclet numbers in the convection-dominated regime. Approximate computations of the residual-free bubbles are performed using a two-level nite element method.