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

Authors: Joseph A. Fiordilino and Ali Pakzad
Journal: Math. Comp. 89 (2020), 629-643
MSC (2010): Primary 65M12, 65M60, 76R10; Secondary 76D05
Published electronically: November 4, 2019
MathSciNet review: 4044444
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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 $ \Vert T^{n}_{h} \Vert $, 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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Additional Information

Joseph A. Fiordilino
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Address at time of publication: Measurement Science and Engineering Department, Naval Surface Warfare Center Corona Division, Corona, California 9287

Ali Pakzad
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405

Received by editor(s): October 6, 2017
Received by editor(s) in revised form: October 18, 2017, and April 22, 2019
Published electronically: November 4, 2019
Additional Notes: The authors’ research was partially supported by NSF grants DMS 1522267 and CBET 1609120
The first author was also supported by the DoD SMART Scholarship
Article copyright: © Copyright 2019 American Mathematical Society