Local projection stabilization methods for $\boldsymbol{H}({\rm curl})$ and $\boldsymbol{H}({\rm div})$ advection problems

TL;DR

This paper introduces local projection stabilization methods for advection problems in H(curl) and H(div) spaces, using finite element spaces with bubble functions to achieve optimal error estimates and verified through numerical tests.

Contribution

It presents a unified framework for LPS methods in H(curl) and H(div) spaces with arbitrary order, utilizing bubble functions for stability and error analysis.

Findings

Optimal a priori error estimates established.

Numerical examples confirm theoretical predictions.

Stabilization effectively improves solution accuracy.

Abstract

We devise local projection stabilization (LPS) methods for advection problems in the $\boldsymbol{H}$(curl) and $\boldsymbol{H}$(div) spaces, employing conforming finite element spaces of arbitrary order within a unified framework. The key ingredient is a local inf-sup condition, enabled by enriching the approximation space with appropriate $\boldsymbol{H}$(d) bubble functions (with d = curl or div). This enrichment allows for the construction of modified interpolation operators, which are crucial for establishing optimal a priori error estimates in the energy norm. Numerical examples are presented to verify both the theoretical results and the stabilization properties of the proposed method.