A Discrete Hopf Interpolant and Stability of the Finite Element Method for Natural Convection

TL;DR

This paper proves that the temperature approximation in finite element methods for natural convection problems can grow at most linearly over time under certain mesh conditions, improving understanding of stability in these simulations.

Contribution

It introduces a discrete Hopf interpolant and establishes a linear-in-time stability result for FEM approximations with nonhomogeneous boundary conditions in natural convection.

Findings

Temperature approximation grows at most linearly in time.

Stability is achieved when the mesh line is within O(Ra^{-1}) of the boundary.

Provides new theoretical bounds for FEM stability in natural convection.

Abstract

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-B\'{e}nard 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.