A note on stable Helmholtz decompositions in 3D

ABSTRACT

The stability of Helmholtz decompositions in 3D is known to hold for convex Lipschitz-continuous polyhedral regions and for arbitrary (not necessarily convex) domains of class $C^{1,1}$. In this note we extend this result to non-convex Lipschitz-continuous polyhedral regions and to the case of homogeneous Neumann boundary conditions on a part of the boundary that is contained in the boundary of a convex extension of the original region. In addition, the relation with very similar results already available in the literature is also discussed. Finally, some implications of our results on the associated discrete Helmholtz decomposition and its application to the derivation of a posteriori error estimates, are briefly described.

KEYWORDS:

• Helmholtz decomposition
• a posteriori error analysis

AMS SUBJECT CLASSIFICATIONS:

• 65N15
• 65N30
• 35J20
• 35J25
• 35Q74
• 65J1

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was partially supported by Comisión Nacional de Investigación Científica y Tecnológica (CONICYT)-Chile through BASAL project CMM, Universidad de Chile, and project Anillo ACT1118 (ANANUM); and by Centro de Investigación en Ingeniería Matemática (CI²MA), Universidad de Concepción.