Robust Preconditioning Estimates for Convection-Dominated Elliptic Problems via a Streamline Poincaré-Friedrichs Inequality

This paper is devoted to the streamline diffusion finite element method, combined with equivalent preconditioning, for solving convection-dominated elliptic problems. The preconditioner is obtained from the streamline diffusion inner product. It is proved that the obtained convergence is robust, i.e., bounded independently of the perturbation parameter $\varepsilon$, for proper convection vector fields. The key to the estimates is an improved “streamline" Poincare--Friedrichs inequality.

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