Decoupled Scheme for Time-Dependent Natural Convection Problem II: Time Semidiscreteness

Numerical Methods Partial

Yuan

2015

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In this article, a decoupled two grid finite element method (FEM) is proposed and analyzed for the nonsteady natural convection problem using the coarse grid numerical solutions to decouple the nonlinear coupled terms, and the corresponding optimal error estimates are derived. Compared with the standard Galerkin FEM and the usual two-grid FEM, our algorithm not only keeps good accuracy but also saves a lot of computational cost. Some numerical examples are provided to verify the performances of the decoupled two-grid FEM. Both theoretical analysis and numerical experiments show the efficiency and effectiveness of the decoupled two-grid FEM for the nonsteady natural convection problem.

Section: Introductionmentioning

confidence: 99%

Decoupled two‐grid finite element method for the time‐dependent natural convection problem I: Spatial discretization

Numerical Methods Partial

Yuan

2015

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“…Theorem Let

f, f_{t} \in L^{\infty} false(0, T_{t i m e}^{f}; Y false)

g, g_{t} \in L^{\infty} false(0, T_{t i m e}^{f}; Z false)

, and u 0 ∈ X , T 0 ∈ W . Suppose the solution of problem satisfies

\begin{matrix} u \in L^{\infty} false(0, T_{t i m e}^{f}; X \cap W^{1, \infty} {false(normalΩ false)}^{2} false) \cap H^{1} false(0, T_{t i m e}^{f}; H^{2} {false(normalΩ false)}^{2} false) \cap H^{2} false(0, T_{t i m e}^{f}; L^{2} {false(normalΩ false)}^{2} false), \\ T \in L^{\infty} false(0, T_{t i m e}^{f}; W \cap W^{1, \infty} false(normalΩ false) false) \cap H^{1} false(0, T_{t i m e}^{f}; H^{2} false(normalΩ false) false) \cap H^{2} false(0, T_{t i m e}^{f}; L^{2} false(normalΩ false) false) . \end{matrix}

Then, system possesses a unique solution ( u n , p n , T n ) and satisfies

\begin{matrix} false‖ u^{J + 1} ‖_{0}^{2} & + normalΔ t ν true \sum_{n = 0}^{J} false‖ \nabla u^{n + 1} ‖_{0}^{2} \leq \frac{2 C_{1}^{4} k ν^{2}}{λ} ((), C_{1}^{2}) \end{matrix}

…”

Section: Galerkin Finite Element Approximationmentioning

confidence: 99%

“…Theorem 3.1. Let , t ∈ L ∞ (0, T time ; Y ), g, g t ∈ L ∞ (0, T time ; Z), and u 0 ∈ X, T 0 ∈ W. 4 Suppose the solution of problem (1) satisfies…”

Section: Galerkin Finite Element Approximationmentioning

confidence: 99%

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

Math Methods in App Sciences

Jiang

Yuan

2018

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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.

“…Under the assumption (A1), for all T > 0 and 0 < t T , the solutions (u, p, θ) of problem (1.1) satisfy (see [16], [19], [22]…”

Section: Preliminariesmentioning

confidence: 99%

“…Generally speaking, it is expensive to find the numerical solutions of the coupled nonlinear system directly by the standard Galerkin FEM. In order to overcome this difficulty, Zhang and his co-authors considered the decoupled schemes for the natural convection problem in [22], [23], [25], and some meaningful results have been established. Recently, Qian and Zhang in [11], [12] considered the first order and higher order projection schemes for the time-dependent natural convection problem.…”

Section: Introductionmentioning

confidence: 99%

The second order projection method in time for the time-dependent natural convection problem

Qian

2016

Appl Math

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