A local Fortin projection for the Scott-Vogelius elements on general meshes

TL;DR

This paper introduces a local Fortin projection for Scott-Vogelius finite elements on general 2D meshes, ensuring divergence preservation, boundary data consistency, and local stability, even with singular vertices.

Contribution

It constructs a novel local Fortin projection for Scott-Vogelius elements on arbitrary shape-regular meshes, including those with singular vertices, with divergence and boundary data preservation.

Findings

Projection preserves divergence in the dual pressure space

Ensures discrete boundary data is maintained

Achieves local stability estimates

Abstract

We construct a local Fortin projection for the Scott-Vogelius finite element pair for polynomial degree $k \ge 4$ on general shape-regular triangulations in two dimensions. In particular, the triangulation may contain singular vertices. In addition to preserving the divergence in the dual of the pressure space, the projection preserves discrete boundary data and satisfies local stability estimates.