Non-Conforming Structure Preserving Finite Element Method for Doubly Diffusive Flows on Bounded Lipschitz Domains
This work addresses numerical simulation challenges in fluid dynamics for porous media and complex geometries, representing an incremental improvement over existing methods.
The authors tackled the problem of modeling doubly diffusive flows with temperature-dependent viscosity on complex domains by developing a non-conforming finite element method that ensures divergence-free velocity fields and unique discrete solutions, achieving theoretical error estimates validated through accuracy and benchmark tests.
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.