Commuting quasi-interpolators and Maxwell compactness for a polytopal de Rham complex

TL;DR

This paper proves Maxwell compactness for the Discrete De Rham polytopal complex using novel quasi-interpolators, enabling convergence analysis of DDR schemes for PDEs with minimal regularity and generic boundary conditions.

Contribution

It introduces new quasi-interpolators that map minimal-regularity de Rham spaces to DDR spaces, forming a commuting diagram and supporting convergence proofs.

Findings

Maxwell compactness results for DDR complex established.

Novel quasi-interpolators with full consistency properties designed.

Convergence proofs for DDR schemes under minimal regularity and various boundary conditions.

Abstract

We establish Maxwell compactness results for the Discrete De Rham (DDR) polytopal complex: sequences in this polytopal complex with bounded discrete $\boldsymbol{H}(\mathbf{curl})$ (resp. discrete $\boldsymbol{H}(\mathrm{div})$) norm and orthogonal to discrete gradients (resp. discrete curls) have $L^2$-relatively compact potential reconstructions. The proof of these results hinges on the design of novel quasi-interpolators, that map the minimal-regularity de Rham spaces onto the discrete DDR spaces and form a commuting diagram. A full set of (primal and adjoint) consistency properties is established for these quasi-interpolators, which paves the way to convergence proofs, under minimal-regularity assumptions, of DDR schemes for partial differential equations based on the de Rham complex. Our analysis is performed with generic mixed boundary conditions, also covering the cases of no boundary conditions or fully homogeneous boundary conditions, and leverages recently introduced liftings from the DDR complex to the continuous de Rham complex.