Parallel two-step finite element algorithm based on fully overlapping domain decomposition for the time-dependent natural convection problem

Abstract

Purpose This paper aims to propose two parallel two-step finite element algorithms based on fully overlapping domain decomposition for solving the 2D/3D time-dependent natural convection problem. Design/methodology/approach The first-order implicit Euler formula and second-order Crank–Nicolson formula are used to time discretization respectively. Each processor of the algorithms computes a stabilized solution in its own global composite mesh in parallel. These algorithms compute a nonlinear system for the velocity, pressure and temperature based on a lower-order element pair (P1b-P1-P1) and solve a linear approximation based on a higher-order element pair (P2-P1-P2) on the same mesh, which shows that the new algorithms have the same convergence rate as the two-step finite element methods. What is more, the stability analysis of the proposed algorithms is derived. Finally, numerical experiments are presented to demonstrate the efficacy and accuracy of the proposed algorithms. Findings Finally, numerical experiments are presented to demonstrate the efficacy and accuracy of the proposed algorithms. Originality/value The novel parallel two-step algorithms for incompressible natural convection problem are proposed. The rigorous analysis of the stability is given for the proposed parallel two-step algorithms. Extensive 2D/3D numerical tests demonstrate that the parallel two-step algorithms can deal with the incompressible natural convection problem for high Rayleigh number well.

A parallel grad‐div stabilized finite element algorithm for the Navier–Stokes equations with a nonlinear damping term

“…Parallel algorithms based on this kind of fully overlapping domain decomposition approach for different problems were studied. [14][15][16][17][18] In literature, there are also another type of parallel algorithms using two-grid discretizations for incompressible flow problems. [19][20][21] To the best knowledge of the authors, all the above mentioned parallel algorithms for incompressible flows are not pressure-robust, that is, the pressure has a strong impact on the approximate velocity.…”

“…Its basic idea is that each processor calculates a local solution in its allocated subdomain using a globally defined mesh, where most of the degrees of freedom comes from the subdomain. Parallel algorithms based on this kind of fully overlapping domain decomposition approach for different problems were studied 14–18 . In literature, there are also another type of parallel algorithms using two‐grid discretizations for incompressible flow problems 19–21 …”

Local and parallel finite element algorithms for the incompressible Navier-Stokes equations with damping

“…Following the idea of the above local and parallel finite element computations, some local and parallel finite element discretization methods were subsequently proposed and investigated for incompressible flow problems. We refer to, for examples, [5,13,15,16,23,24,[27][28][29][30][31]33,36,39] for details. It is shown by numerical results that these local and parallel finite element methods are highly efficient.…”