Statically Condensed Iterated Penalty Method for High Order Finite Element Discretizations of Incompressible Flow

Abstract

We introduce and analyze a Statically Condensed Iterated Penalty (SCIP) method for solving incompressible flow problems discretized with $p$th-order Scott-Vogelius elements. While the standard iterated penalty method is often the preferred algorithm for computing the discrete solution, it requires inverting a linear system with $\mathcal{O}(p^d)$ unknowns at each iteration. The SCIP method reduces the size of this system to $\mathcal{O}(p^{d-1})$ unknowns while maintaining the geometric rate of convergence of the iterated penalty method. The application of SCIP to Kovasznay flow and Moffatt eddies shows good agreement with the theory.

25 pages, 7 figures