A Discrete Hopf Interpolant and Stability of the Finite Element Method for Natural Convection
For researchers in computational fluid dynamics, this provides a theoretical guarantee of boundedness for FEM solutions of natural convection, addressing a long-standing gap in stability analysis.
The paper proves that the finite element method for natural convection problems with nonhomogeneous Dirichlet boundary conditions yields a temperature approximation that grows at most linearly in time, under the condition that the first mesh line is within O(Ra^{-1}) of the boundary. This improves on previous results that only showed stability with possible exponential growth.
The temperature in natural convection problems is, under mild data assumptions, uniformly bounded in time. This property has not yet been proven for the standard finite element method (FEM) approximation of natural convection problems with nonhomogeneous partitioned Dirichlet boundary conditions, e.g., the differentially heated vertical wall and Rayleigh-Bénard problems. For these problems, only stability in time, allowing for possible exponential growth of $\| T^{n}_{h} \| $, has been proven using Gronwall's inequality. Herein, we prove that the temperature approximation can grow at most linearly in time provided that the first mesh line in the finite element mesh is within $\mathcal{O} (Ra^{-1})$ of the nonhomogeneous Dirichlet boundary.