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The Grad-Div Conforming Virtual Element Method for the Quad-Div Problem in Three Dimensions
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DOI：10.1007/s10915-026-03429-y.png)
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Abstract
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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 $$\varvec{H}(\operatorname {grad-div})$$ -conforming virtual element method of arbitrary approximation orders on polyhedral meshes. Three families of $$\varvec{H}(\operatorname {grad-div})$$ -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 on each face. The rigorous analysis includes interpolation error estimates, stability of the discrete bilinear forms, well-posedness of the discrete formulation, and optimal convergence rates. Some numerical examples are presented to verify the theoretical results.
Keywords:
\(\varvec{H}(\operatorname {grad-div})\)-conforming
Virtual elements
Quad-div problem
Variational formulation
de Rham complex
Polyhedral meshes
Journal
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3.3
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652
Citations:
9.6K
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