The grad-div conforming virtual element method for the quad-div problem in three dimensions

TL;DR

This paper introduces a new stable variational formulation and an $oldsymbol{H}( ext{grad-div})$-conforming virtual element method for the 3D quad-div problem, with rigorous analysis and numerical verification.

Contribution

It develops a novel $oldsymbol{H}( ext{grad-div})$-conforming virtual element method with an exact discrete complex for the 3D quad-div problem, including error analysis and stability proofs.

Findings

Proved well-posedness of the new formulation

Established optimal error estimates for the method

Numerical examples confirm theoretical results

Abstract

We propose a new stable variational formulation for the quad-div problem in three dimensions and prove its well-posedness. Using this weak form, we develop and analyze the $\boldsymbol{H}(\operatorname{grad-div})$-conforming virtual element method of arbitrary approximation orders on polyhedral meshes. Three families of $\boldsymbol{H}(\operatorname{grad-div})$-conforming virtual elements are constructed based on the structure of a de Rham sub-complex with enhanced smoothness, resulting in an exact discrete virtual element complex. In the lowest-order case, the simplest element has only one degree of freedom at each vertex and face, respectively. We rigorously prove the interpolation error estimates, the stability of discrete bilinear forms, the well-posedness of discrete formulation and the optimal error estimates. Some numerical examples are shown to verify the theoretical results.