Optimal Error Analysis of Linearized Crank-Nicolson Finite Element Scheme for the Time-Dependent Penetrative Convection Problem

Abstract

This paper focuses on the optimal error analysis of a linearized Crank-Nicolson finite element scheme for the time-dependent penetrative convection problem, where the mini element and piecewise linear finite element are used to approximate the velocity field, the pressure, and the temperature, respectively. We proved that the proposed finite element scheme is unconditionally stable and the optimal error estimates in $L^2$-norm are derived. Finally, numerical results are presented to confirm the theoretical analysis.

Keywords

Time-dependent penetrative convection problem / Linearized Crank-Nicolson scheme / Finite element method / Error estimate / 36Q30 / 76M10 / 65M60

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Funding

National Natural Science Foundation of China(11771337)

Natural Science Foundation of Zhejiang Province(LY23A010002)

RIGHTS & PERMISSIONS

Shanghai University