Discontinuous Galerkin methods for the Stokes equations with nonlinear damping term on general meshes

This paper introduces a three-step Oseen-linearized finite element method for the 2D/3D steady incompressible Navier-Stokes equations with nonlinear damping term. Within this method, we solve a nonlinear problem over a coarse grid followed by solving two Oseen-linearized problems over a fine grid, which possess the same stiffness matrices with only various right-hand sides. We theoretically analyze the stability of the present method, and derive optimal error estimates of the finite element solutions. We conduct a series of numerical experiments which support the theoretical analysis and test the effectiveness of the proposed method. We demonstrate numerically that there is a significant improvement in the accuracy of the approximate solutions over those for the standard two-level method.

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“…For problem (15), we have the following results (see, e.g., Theorems 2-4 on pages 296-300 of [26] and lemma 5 on page 439 of [11] for details).…”

Section: Preliminariesmentioning

confidence: 99%

A three‐step Oseen‐linearized finite element method for incompressible flows with damping

Zheng

Shang

2022

Numerical Methods Partial

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“…5 Simultaneously, recent studies have paid more attention to devising some efficiently numerical approaches based on finite element approximations for the Stokes and Navier-Stokes equations with nonlinear damping term. We can refer to previous works [7][8][9][10][11][12][13][14][15][16][17] and the references therein. In the above methods, finite element methods are quite interesting from researchers in that the methods in approximating the solution domain are flexible, and their developments of theoretical analysis are perfect.…”

Section: Introductionmentioning

confidence: 99%

Two‐grid stabilized algorithms for the steady Navier–Stokes equations with damping

Zheng

Shang

2022

Math Methods in App Sciences

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This paper presents and studies three two-grid stabilized quadratic equal-order finite element algorithms based on two local Gauss integrations for the steady Navier-Stokes equations with damping. In these algorithms, we first solve a stabilized nonlinear problem on a coarse grid, and then pass the coarse grid solution to a fine grid and solve a stabilized linear problem. Using some nonlinear analysis techniques, we analyze stability of the algorithms and derive optimal order error estimates of the approximate solutions. Theoretical and numerical results show that, when the algorithmic parameters are chosen appropriately, the accuracy of the approximate solutions computed by our two-grid stabilized algorithms is comparable to that of solving a fully stabilized nonlinear problem on the same fine grid; however, our two-grid algorithms save a large amount of CPU time than the one-grid stabilized algorithm.

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“…In a discontinuous Galerkin finite element method, 7 the velocity u is also approximated by piecewise polynomials of degree k but discontinuous, and the pressure p by discontinuous piecewise polynomials of degree k − 1 (or k ) on polygonal/polyhedral meshes, with the following formulation: Find

u_{h} \in V_{h} \subset L^{2} (Ω)

and

p_{h} \in W_{h} \subset L_{0}^{2} (Ω)

such that

\begin{align} \sum_{T \in 𝒯_{h}} ([], {false(normal∇ {boldu}_{h}, normal∇ {boldv}_{h} false)}_{T} prefix- {false(p_{h}, normal∇ \cdot {boldv}_{h} false)}_{T}) \\ + \sum_{e \in ℰ_{h}} ((, \int_{e} false{normal∇ {boldu}_{h} false} boldn \cdot false[{boldv}_{h} false] + ϵ^{*} \int_{e} false{normal∇ {boldv}_{h} false} boldn \cdot false[{boldu}_{h} false]) \\ (), + \int_{e} false{p_{h} false} boldn \cdot false[{boldv}_{h} false] + \frac{σ_{e}}{h} \int_{e} false[{boldu}_{h} false] \cdot false[{boldv}_{h} false]) = false(boldf, \end{align}

…”

Section: Introductionmentioning

confidence: 99%

A conforming discontinuous Galerkin finite element method for the Stokes problem on polytopal meshes

Zhang

2021

Numerical Methods in Fluids

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A new discontinuous Galerkin finite element method for the Stokes equations is developed in the primary velocity‐pressure formulation. This method employs discontinuous polynomials for both velocity and pressure on general polygonal/polyhedral meshes. Most finite element methods with discontinuous approximation have one or more stabilizing terms for velocity and for pressure to guarantee stability and convergence. This new finite element method has the standard conforming finite element formulation, without any velocity or pressure stabilizers. Optimal‐order error estimates are established for the corresponding numerical approximation in various norms. The numerical examples are tested for low and high order elements up to the degree four in 2D and 3D spaces.

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Discontinuous Galerkin methods for the Stokes equations with nonlinear damping term on general meshes

Cited by 13 publications

References 30 publications

A three‐step Oseen‐linearized finite element method for incompressible flows with damping

A three‐step Oseen‐linearized finite element method for incompressible flows with damping

Two‐grid stabilized algorithms for the steady Navier–Stokes equations with damping

A conforming discontinuous Galerkin finite element method for the Stokes problem on polytopal meshes

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