Discontinuous Galerkin Method with Staggered Hybridization for a Class of Nonlinear Stokes Equations

Du, Jian; Chung, Eric T.; Lam, Ming Fai; Wang, Xiaoping

doi:10.1007/s10915-018-0676-z

Cited by 10 publications

(5 citation statements)

References 25 publications

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“…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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Section: Introductionmentioning

confidence: 99%

Mortar DG Method with Staggered Hybridization for Rayleigh Waves Simulation

Chung¹

2021

CICP

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“…Note that SDG method differs from other DG methods in the sense that the basis functions are locally conforming and penalty terms are not required. This method has many desirable features and has been successfully applied to a wide range of partial differential equations [9,21,52,56,53,54,51,55,31].…”

Section: Introductionmentioning

confidence: 99%

A pressure robust staggered discontinuous Galerkin method for the Stokes equations

Zhao

Park²,

Chung

2020

Preprint

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In this paper we propose a pressure robust staggered discontinuous Galerkin method for the Stokes equations on general polygonal meshes by using piecewise constant approximations. We modify the right hand side of the body force in the discrete formulation by exploiting divergence preserving velocity reconstruction operator, which is the crux for pressure independent velocity error estimates. The optimal convergence for velocity gradient, velocity and pressure are proved. In addition, we are able to prove the superconvergence of velocity approximation by the incorporation of divergence preserving velocity reconstruction operator in the dual problem, which is also an important contribution of this paper. Finally, several numerical experiments are carried out to confirm the theoretical findings.

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“…Staggered discontinuous Galerkin (SDG) methods pioneered by Chung and Engquist [13,14] earn many desirable properties such as mass conservation, superconvergence and the flexibility to deal with general quadrilateral and polygonal meshes, and have been applied to numerous partial differential equations arising from the practical applications (cf. [15,28,17,20,16,12,38,40]). To further advance the applications of SDG method, a mortar formulation is developed for SDG method [24], where different triangulations in different regions of the computational domain are exploited.…”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimates for the mortar staggered DG method

Zhao,

Chung

2019

Preprint

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Two residual-type error estimators for the mortar staggered discontinuous Galerkin discretizations of second order elliptic equations are developed. Both error estimators are proved to be reliable and efficient. Key to the derivation of the error estimator in potential L 2 error is the duality argument. On the other hand, an auxiliary function is defined, making it capable of decomposing the energy error into conforming part and nonconforming part, which can be combined with the well-known Scott-Zhang local quasi-interpolation operator and the mortar discrete formulation yields an error estimator in energy error. Importantly, our analysis for both error estimators does not require any saturation assumptions which are often needed in the literature. Several numerical experiments are presented to confirm our proposed theories.

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Discontinuous Galerkin Method with Staggered Hybridization for a Class of Nonlinear Stokes Equations

Cited by 10 publications

References 25 publications

Mortar DG Method with Staggered Hybridization for Rayleigh Waves Simulation

Mortar DG Method with Staggered Hybridization for Rayleigh Waves Simulation

A pressure robust staggered discontinuous Galerkin method for the Stokes equations

A posteriori error estimates for the mortar staggered DG method

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