A Hybrid Two-Level Finite Element Method for the Stationary Natural Convection Problem and Parallel Implement

Abstract

This study presents a novel hybrid approach for addressing incompressible stationary natural convection problem, incorporating a parallel technique to enhance computational efficiency. Inspired by the traditional two-level method [He and Wang, Comput. Methods Appl. Mech. Engrg., 197 (2008)] and the two-step approach [Wu et al., Int. J. Heat Mass Transfer, 101 (2016)], both characterized by their iterative and corrective processes, we endeavor to alleviate the computational burden associated with the iterative process. Building upon these methods, our novel hybrid method involves two primary steps: initially solving the original problem using the finite element pair $P_{1}b-P_1-P_1$ on a coarse mesh, followed by resolving the linearized equations using the higher-order pair $P_2-P_1-P_2$ on a fine mesh. While the first step employs iterative techniques, the second step entails directly solving a linearized problem. This novel approach can save lots of computational time in the iterative step compared to the traditional methods. Moreover, leveraging domain decomposition techniques, we implement a parallel strategy to further accelerate computations. Finally, we conduct several numerical examples to validate the efficiency of the proposed algorithms. The numerical results demonstrate optimal convergence rates comparable to those obtained using only the $P_2-P_1-P_2$ finite element pair under similar relative error conditions. Furthermore, the numerical simulations on the two obstacles flow and Bénard convection problem show the robustness and efficiency of the proposed algorithms.