Non-Conforming Structure Preserving Finite Element Method for Doubly Diffusive Flows on Bounded Lipschitz Domains

TL;DR

This paper introduces a new non-conforming finite element method for modeling doubly diffusive flows with temperature-dependent viscosity, ensuring divergence-free velocity fields and establishing theoretical and numerical validation.

Contribution

It presents a novel finite element scheme based on Crouzeix-Raviart elements that guarantees unique discrete solutions and provides rigorous error analysis for complex flow regimes.

Findings

The method achieves optimal error decay rates in various flow regimes.

It ensures locally exactly divergence-free velocity fields.

Numerical tests confirm theoretical error estimates and robustness across domain types.

Abstract

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.