Susanne C. Brenner, Li-yeng Sung, Jai Tushar
We design and analyze virtual element methods for a quad-curl problem on general polygonal domains that are based on the Hodge decomposition of divergence-free vector fields. Numerical results that corroborate the theoretical analysis are also presented.
Jai Tushar, Arbaz Khan, Manil T. Mohan
We study a stationary model of doubly diffusive flows with temperature-dependent viscosity on bounded Lipschitz domains in two and three dimensions. A new well-posedness and regularity analysis of weak solutions under minimal assumptions on domain geometry and data regularity are established. A fully non-conforming finite element method based on Crouzeix-Raviart elements, which ensures locally exactly divergence-free velocity fields is explored. Unlike previously proposed schemes, this discretization enables to establish uniqueness of the discrete solutions. We prove the well-posedness of the discrete problem and derive a priori error estimates. An accuracy test is conducted to verify the theoretical error decay rates in flow, Stokes and Darcy regimes on convex and non-convex domains, and a benchmark test of flow in a porous cavity is conducted, comparing the proposed method with existing literature.