An efficient two-step algorithm for steady-state natural convection problem

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

doi:10.1016/j.ijheatmasstransfer.2016.05.061

Cited by 30 publications

(23 citation statements)

References 32 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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“…By the inf-sup condition (9) with the first equation in (15), one finds (21), Theorems 3.6 and 3.7 and the triangle inequality, we finish the proof.…”

Section: Lemma 35 Under the Conditions Of Lemma 34 There Holdmentioning

confidence: 55%

“…This kind of discretization technique for linear and nonlinear elliptic equations was first introduced by Xu. 9,10 After then, two-level method has been studied by many researchers, for example, we can refer to He, [11][12][13] Layton and Lenferink, 14 and Wu et al 15 for the incompressible flow. In this paper, we consider the two-level method for the Boussinesq problem.…”

Section: Introductionmentioning

confidence: 99%

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One‐level and two‐level finite element methods for the Boussinesq problem

Zhang

Jiang

Yuan

2018

Math Methods in App Sciences

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Previous works on the convergence of numerical methods for the Boussinesq problem were conducted, while the optimal L2‐norm error estimates for the velocity and temperature are still lacked. In this paper, the backward Euler scheme is used to discrete the time terms, standard Galerkin finite element method is adopted to approximate the variables. The MINI element is used to approximate the velocity and pressure, the temperature field is simulated by the linear polynomial. Under some restriction on the time step, we firstly present the optimal L2 error estimates of approximate solutions. Secondly, two‐level method based on Stokes iteration for the Boussinesq problem is developed and the corresponding convergence results are presented. By this method, the original problem is decoupled into two small linear subproblems. Compared with the standard Galerkin method, the two‐level method not only keeps good accuracy but also saves a lot of computational cost. Finally, some numerical examples are provided to support the established theoretical analysis.

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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 steady-state natural convection problem

Cited by 30 publications

References 32 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

One‐level and two‐level finite element methods for the Boussinesq problem

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

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