A new staggered DG method for the Brinkman problem robust in the Darcy and Stokes limits

Zhao, Lina; Chung, Eric T.; Lam, Ming Fai

doi:10.1016/j.cma.2020.112986

Cited by 28 publications

(20 citation statements)

References 41 publications

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“…In this study, we developed a high-order polygonal staggered DG method for the Navier-Stokes equations. A new finite element pair is introduced in the spirit of [47], which enables us to achieve pressure robustness. It is worth mentioning that the proposed scheme yields a divergence free velocity approximation without any postprocessing which is a desirable feature in the applications of (Navier-)Stokes equations.…”

Section: Discussionmentioning

confidence: 99%

“…We can define the following degrees of freedom in the spirit of [15,47]. (XD3) For each τ ∈ T h , we can obtain…”

Section: Staggered Discontinuous Galerkin Methodsmentioning

confidence: 99%

“…Recently, Zhao and Park [49] extended this method to general polygonal meshes for the Poisson equation. Then, a high order staggered DG method for general secondorder elliptic problems is developed in [51], and it is applied to various physical problems arising from practical applications [55,51,36,52,47,50,54,48]. For a pressure-robust method, Zhao et al [53] proposed a lowest-order pressure-robust staggered DG method for the Stokes equations.…”

Section: Introductionmentioning

confidence: 99%

“…The objective of this paper is to design and analyze a pressure robust staggered DG method of arbitrary polynomial orders on polygonal meshes for the Navier-Stokes problem. The discrete formulation involves velocity gradient, velocity and pressure, and the continuities for all the variables are staggered on the inter-element boundaries in line with [47]. The staggered continuity naturally gives an inter-element flux term which is free of stabilization parameters and meanwhile ensures the inf-sup stability of the resulting bilinear forms.…”

Section: Introductionmentioning

confidence: 99%

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Pressure-robust staggered DG methods for the Navier-Stokes equations on general meshes

Kim,

Zhao,

Chung

et al. 2021

Preprint

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In this paper, we design and analyze staggered discontinuous Galerkin methods of arbitrary polynomial orders for the stationary Navier-Stokes equations on polygonal meshes. The exact divergence-free condition for the velocity is satisfied without any postprocessing. The resulting method is pressure-robust so that the pressure approximation does not influence the velocity approximation. A new nonlinear convective term that earning nonnegativity is proposed. The optimal convergence estimates for all the variables in L 2 norm are proved. Also, assuming that the rotational part of the forcing term is small enough, we are able to prove that the velocity error is independent of the Reynolds number and of the pressure. Furthermore, superconvergence can be achieved for velocity under a suitable projection. Numerical experiments are provided to validate the theoretical findings and demonstrate the performances of the proposed method.

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Section: Discussionmentioning

confidence: 99%

“…We can define the following degrees of freedom in the spirit of [15,47]. (XD3) For each τ ∈ T h , we can obtain…”

Section: Staggered Discontinuous Galerkin Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Pressure-robust staggered DG methods for the Navier-Stokes equations on general meshes

Kim,

Zhao,

Chung

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Motivated by the staggered grid finite difference schemes, the staggered discontinuous Galerkin (SDG) method is developed with the goal of achieving high order accuracy on domains with irregular geometries and keeping the advantages of using staggered meshes. The SDG methods are successfully applied to both the acoustic wave equations [7,8,20] and the elastic wave equations [6,9] as well as other applications [3,17,31]. The use of staggered mesh in discontinuous Galerkin method offers several additional advantages such as energy conservation, optimal rate of convergence and low dispersion error [2,9].…”

Section: Introductionmentioning

confidence: 99%

Mortar DG Method with Staggered Hybridization for Rayleigh Waves Simulation

Chung¹

2021

CICP

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The simulation of Rayleigh waves is important in a variety of geophysical applications. The computational challenge is the fact that very fine mesh is necessary as the waves are concentrated at the free surface and decay exponentially away from the free surface. To overcome this challenge and to develop a robust high order scheme for the simulation of Rayleigh waves, we develop a mortar discontinuous Galerkin method with staggered hybridization. The use of the mortar technique allows one to use fine mesh in only a local region near the free surface, and use coarse mesh in most of the domain. This approach reduces the computational cost significantly. The staggered hybridization allows the preservation of the strong symmetry of the stress tensor without complicated construction of basis functions. In particular, the basis functions are piecewise polynomial without any continuity requirement, and the coupling of the basis functions is performed by using carefully chosen hybridized variables. The resulting scheme is explicit in time, and only local saddle point system are solved for each time step. We will present several benchmark problems to demonstrate the performance of the proposed method.

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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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A new staggered DG method for the Brinkman problem robust in the Darcy and Stokes limits

Cited by 28 publications

References 41 publications

Pressure-robust staggered DG methods for the Navier-Stokes equations on general meshes

Pressure-robust staggered DG methods for the Navier-Stokes equations on general meshes

Mortar DG Method with Staggered Hybridization for Rayleigh Waves Simulation

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

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