A grad-curl conforming virtual element method for a grad-curl problem linking the 3D quad-curl problem and Stokes system
Year: 2025
Type: preprint
Abstract: Based on the Stokes complex with vanishing boundary conditions and its dual complex, we reinterpret a grad-curl problem arising from the quad-curl problem as a new vector potential formulation of the three-dimensional Stokes system. By extending the analysis to the corresponding non-homogeneous problems and the accompanying trace complex, we construct a novel $\boldsymbol{H}(\operatorname{grad-curl})$-conforming virtual element space with arbitrary approximation order that satisfies the exactness of the associated discrete Stokes complex. In the lowest-order case, three degrees of freedom are assigned to each vertex and one to each edge. For the grad-curl problem, we rigorously establish the interpolation error estimates, the stability of discrete bilinear forms, and the convergence of the proposed element on polyhedral meshes. As a discrete vector potential formulation of the Stokes problem, the resulting system is pressure-decoupled and symmetric positive definite. Some numerical examples are presented to verify the theoretical results.
Source: arXiv (Cornell University)
Authors: Dong, Xiaojing, Han, Yibing, Huang, Yunqing
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7101680040
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