A discrete Hopf interpolant and stability of the finite element method for natural convection

Fiordilino, Joseph; Pakzad, Ali

doi:10.1090/mcom/3489

A discrete Hopf interpolant and stability of the finite element method for natural convection
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by Joseph A. Fiordilino and Ali Pakzad HTML | PDF

Math. Comp. 89 (2020), 629-643 Request permission

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–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.

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