Newton iterative method based on finite element discretization for the stationary Darcy–Brinkman equations

Wang, Zihao; Huang, Pengzhan; Wang, Kun

doi:10.1016/j.camwa.2020.10.020

Cited by 4 publications

(12 citation statements)

References 27 publications

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“…([ 7 , 8 , 26 , 27 ]) Let

. Under the assumption of Theorem 2, the numerical solution pair

to ( 12 ) satisfies

and

Moreover, the following error estimate holds

where c is a positive constant depending on h .…”

Section: Several Iterative Methods Based On the Finite Element Discre...mentioning

confidence: 99%

“…Assume u 1 , u 2 ∈ X and subtract the equations obtained from (7) with u = u 1 and u = u 2 . Then, set w = A(u 2 ) − A(u 1 ) and choose v = w to yield…”

Section: Existence and Uniquenessmentioning

confidence: 99%

“…In this numerical experiment, we assume that the boundary conditions satisfy [ 7 , 9 ]

and set

and the mesh size

. Moreover, the convergence tolerance is set to equal

.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Iterative method I and II are valid as σ ∈ [ 1 4 , 1 3 ) and only iterative method I runs well as σ ∈ [ 1 3 , 1). In this paper, by developing some techniques and using some ideas in [7], we prove the existence and uniqueness with a different method, then we obtain a different uniqueness condition. Furthermore, we propose and analyze iterative methods I and III.…”

Section: Introductionmentioning

confidence: 99%

“…The governing equations of this double-diffusive natural convection model are presented as follows [ 7 ].

…”

Section: Introductionmentioning

confidence: 99%

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Finite Element Iterative Methods for the Stationary Double-Diffusive Natural Convection Model

Wei

Huang

2022

Entropy

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In this paper, we consider the stationary double-diffusive natural convection model, which can model heat and mass transfer phenomena. Based on the fixed point theorem, the existence and uniqueness of the considered model are proved. Moreover, we design three finite element iterative methods for the considered problem. Under the uniqueness condition of a weak solution, iterative method I is stable. Compared with iterative method I, iterative method II is stable with a stronger condition. Moreover, iterative method III is stable with the strongest condition. From the perspective of viscosity, iterative method I displays well in the case of a low viscosity number, iterative method II runs well with slightly low viscosity, and iterative method III can deal with high viscosity. Finally, some numerical experiments are presented for testing the correctness of the theoretic analysis.

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“…([ 7 , 8 , 26 , 27 ]) Let

. Under the assumption of Theorem 2, the numerical solution pair

to ( 12 ) satisfies

and

Moreover, the following error estimate holds

where c is a positive constant depending on h .…”

Section: Several Iterative Methods Based On the Finite Element Discre...mentioning

confidence: 99%

“…Assume u 1 , u 2 ∈ X and subtract the equations obtained from (7) with u = u 1 and u = u 2 . Then, set w = A(u 2 ) − A(u 1 ) and choose v = w to yield…”

Section: Existence and Uniquenessmentioning

confidence: 99%

“…In this numerical experiment, we assume that the boundary conditions satisfy [ 7 , 9 ]

and set

and the mesh size

. Moreover, the convergence tolerance is set to equal

.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The governing equations of this double-diffusive natural convection model are presented as follows [ 7 ].

…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Finite Element Iterative Methods for the Stationary Double-Diffusive Natural Convection Model

Wei

Huang

2022

Entropy

Self Cite

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High‐performance computational homogenization of Stokes–Brinkman flow with an Anderson‐accelerated FFT method

Chen

2023

Numerical Methods in Fluids

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Fluid flow problems in bi‐porous media have strong implications in a variety of industrial applications, such as nuclear waste disposal and composites manufacturing. Despite various numerical solutions proposed in the literature, the prediction of dual‐scale flow remains a challenging task due to the requirement of fine meshes to resolve the complex microstructures of bi‐porous media. This paper introduces a new numerical solver based on fast Fourier transform (FFT) for the Stokes–Brinkman problem to predict fluid flow in bi‐porous media with local anisotropy. The novel FFT solver is derived from a velocity‐form of the problem, in contrast to the stress‐form proposed by a recent work. The Anderson acceleration technique is applied for the first time to the Stokes–Brinkman problem, leading to a substantial (orders of magnitude) improvement of the convergence rate. A range of detailed numerical examples are provided to validate the method against the analytical and literature results. Parametric studies are also demonstrated to aid in the selection of model parameters to achieve optimum numerical performance. With a 3D unit cell of a woven textile fabric, we demonstrate that the proposed method is capable of handling high‐resolution simulations with strongly heterogeneous microstructures. Combined with a parallelized implementation over high‐performance computing systems, our method demonstrates a new state‐of‐the‐art in numerical solvers for the Stokes‐Brinkman problem, in terms of computation capacity.

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Optimal error estimates of two‐level iterative finite element methods for the thermally coupled incompressible MHD with different viscosities

Zhang

2022

Math Methods in App Sciences

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In this paper, we develop and design the two‐level finite element iterative methods for the stationary thermally coupled incompressible MHD equations. The considered numerical schemes are based on the classical iterations with one correction. First, under some strong uniqueness conditions, the rough solutions are obtained on the coarse mesh with the mesh size and the Simple, Oseen, and Newton iterations, respectively. Three kinds of corrections are made by solving a linear problem on the fine mesh with mesh size with different viscosities. Finally, under the weak uniqueness condition, the stationary thermally coupled incompressible MHD is solved by the one‐level finite element Oseen iteration on the fine mesh. The uniform stability and convergence of these two‐level iterative methods are analyzed with respect to the mesh sizes and iterative times . Extensive numerical results are presented to demonstrate the established theoretical findings and show the performances of these two‐level iterative schemes with different viscosities.

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Newton iterative method based on finite element discretization for the stationary Darcy–Brinkman equations

Cited by 4 publications

References 27 publications

Finite Element Iterative Methods for the Stationary Double-Diffusive Natural Convection Model

Finite Element Iterative Methods for the Stationary Double-Diffusive Natural Convection Model

High‐performance computational homogenization of Stokes–Brinkman flow with an Anderson‐accelerated FFT method

Optimal error estimates of two‐level iterative finite element methods for the thermally coupled incompressible MHD with different viscosities

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