An efficient two-step algorithm for the stationary incompressible magnetohydrodynamic equations

Wu, Jilian; Liu, Demin; Feng, Xinlong; Huang, Pengzhan

doi:10.1016/j.amc.2017.01.005

Cited by 11 publications

(10 citation statements)

References 21 publications

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“…In fact, the two‐step method can be cast in the framework of the two‐level method, but compared with the two‐level method, the two‐step method only needs one mesh size, which can avoid the discussion on the relation of the coarse and fine meshes. The two‐step method has already been applied to the steady‐state natural convection equations, the stationary magnetohydrodynamics (MHD), and the Navier‐Stokes equations with friction boundary conditions …”

Section: Introductionmentioning

confidence: 99%

Stabilized mixed finite element methods for the Navier‐Stokes equations with damping

Shi

2018

Math Methods in App Sciences

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In this paper, the stabilized mixed finite element methods are presented for the Navier‐Stokes equations with damping. The existence and uniqueness of the weak solutions are proven by use of the Brouwer fixed‐point theorem. Then, optimal error estimates for the H1‐norm and L2‐norm of the velocity and the L2‐norm of the pressure are derived. Moreover, on the basis of the optimal L2‐norm error estimate of the velocity, a stabilized two‐step method is proposed, which is more efficient than the usual stabilized methods. Finally, two numerical examples are implemented to confirm the theoretical analysis.

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Section: Introductionmentioning

confidence: 99%

Stabilized mixed finite element methods for the Navier‐Stokes equations with damping

Shi

2018

Math Methods in App Sciences

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“…Lemma 3.1 (Wu et al , 2017) We denote δ as difference of 2 consecutive functions, for example, for any sequence {An}n=0N: then we have: and …”

Section: Parallel Two-step Algorithmmentioning

confidence: 99%

“…Natural convection problem exists in fluid flow and heat transfer and has a great application in many industries such as nuclear reactor, animal blood flow, colloidal suspension, solar collector and turbulent shear flow (Su et al , 2017; Su et al , 2014; Wang et al , 2019; Wang et al , 2018; Wu et al , 2017; Zhang et al , 2018; Wang et al , 2019; Szumbarski et al , 2014; Yapici and Obut, 2015; Davis, 1983; Sheremet et al , 2017).…”

Section: Introductionmentioning

confidence: 99%

“…At the same time, many authors also studied the problem of non-stationary natural convection. Luo et al (2003) proposed a hybrid FEM based on finite difference format to handle large Rayleigh cases Wu et al (2017). gave pressure-correction projection FEM to solve the non-stationary natural convection problem and presented (Wu et al , 2015b) the characteristic variational multi-scale method to simulate time-dependent conduction-convection problem.…”

Section: Introductionmentioning

confidence: 99%

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

Ping

Zhao

et al. 2019

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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.

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“…In fact, two-space method [27][28][29] can be cast in the framework of Xu's work regarding the two-grid method. However, the two-space method is different from the twogrid method.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Algorithm with Stabilized Finite Element Method for the Stokes Eigenvalue Problem

Weng

Cai

2017

Mathematical Problems in Engineering

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This paper provides a two-space stabilized mixed finite element scheme for the Stokes eigenvalue problem based on local Gauss integration. The two-space strategy contains solving one Stokes eigenvalue problem using the 1 − 1 finite element pair and then solving an additional Stokes problem using the 2 − 2 finite element pair. The postprocessing technique which increases the order of mixed finite element space by using the same mesh can accelerate the convergence rate of the eigenpair approximations. Moreover, our method can save a large amount of computational time and the corresponding convergence analysis is given. Finally, numerical results are presented to confirm the theoretical analysis.

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An efficient two-step algorithm for the stationary incompressible magnetohydrodynamic equations

Cited by 11 publications

References 21 publications

Stabilized mixed finite element methods for the Navier‐Stokes equations with damping

Stabilized mixed finite element methods for the Navier‐Stokes equations with damping

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

An Efficient Algorithm with Stabilized Finite Element Method for the Stokes Eigenvalue Problem

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