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

Abstract

We devise local projection stabilization (LPS) methods for advection problems in the $\mathbf{H}$(curl) and $\mathbf{H}$(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 $\mathbf{H}$(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.