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

TL;DR

This paper presents a new Statically Condensed Iterated Penalty (SCIP) method that efficiently solves high-order finite element discretizations of incompressible flow, reducing computational complexity while preserving convergence rates.

Contribution

The paper introduces the SCIP method, which reduces the size of linear systems in high-order finite element discretizations of incompressible flow, maintaining convergence efficiency.

Findings

SCIP reduces system size from O(p^d) to O(p^{d-1}) unknowns.

SCIP maintains the geometric convergence rate of the standard iterated penalty method.

Numerical experiments on Kovasznay flow and Moffatt eddies validate the theoretical results.

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.