An analysis on the penalty and Nitsche's methods for the Stokes–Darcy system with a curved interface

Zhou, Guanyu; Kashiwabara, Takahito; Oikawa, Issei; Chung, Eric T.; Shiue, Ming-Cheng

doi:10.1016/j.apnum.2021.02.006

Cited by 10 publications

(4 citation statements)

References 42 publications

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“…Various numerical discretizations were developed subsequently. The discontinuous Galerkin method was considered in References 8,9 and 13. The mortar element method was proposed in References 1 and 4.…”

Section: Introductionmentioning

confidence: 99%

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An extended finite element method for coupled Darcy–Stokes problems

Cao

Chen

2022

Numerical Meth Engineering

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In this article, we propose an extended finite element method to deal with coupled Darcy-Stokes problems with straight or curved interfaces. A mixed approximate form is presented based on Brezzi-Douglas-Marini finite element space and piecewise constant function space for Darcy equations, and a mixed approximate form is shown based on piecewise quadratic function space and piecewise constant function space for Stokes equations. By introducing some stabilization terms, we prove that the inf-sup condition for the discrete coupled problem is valid. Moreover, the optimal convergence is derived. Finally, some numerical examples are presented to demonstrate our theoretical results.

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Section: Introductionmentioning

confidence: 99%

“…However, it is more realistic to model the coupled problem with curved interfaces. As far as we know, there is one paper 13 to deal with the Darcy–Stokes system with a curved interface based on interface‐fitted meshes. But for complicated or moving interface, the generation of interface‐fitted meshes is expensive.…”

Section: Introductionmentioning

confidence: 99%

An extended finite element method for coupled Darcy–Stokes problems

Cao

Chen

2022

Numerical Meth Engineering

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“…The penalty method is effective in the numerical solution of constrained problems. It is a useful tool in proving the existence of the solution to constrained problems, see ( [20]- [22]). And the generalized penalty method is the generalization of the penalty method, cf.…”

Section: Introductionmentioning

confidence: 99%

Generalized Penalty Method for a Class of Variational-hemivariational Inequality

Liu¹

2022

JMI

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“…As it is well known, the standard mixed formulations for the Stokes equations and Darcy equations earn different compatibility conditions, thus a straightforward application of the existing solvers for the Stokes equations and Darcy equations may not be feasible. To this end, a great amount of effort has been devoted to developing accurate and efficient numerical schemes for the coupled Stokes-Darcy problem, and a non-exhaustive list of these approaches include Lagrange multiplier methods [21,17,32,18], weak Galerkin method [9,22], strongly conservative methods [20,16], stabilized mixed finite element method [28,24], discontinuous Galerkin (DG) methods [26,34], virtual element method [23,33], a lowest-order staggered DG method [37] and penalty methods [38]. The coupled Stokes-Darcy problem describes multiphysics phenomena, and involves a Stokes subproblem and a Darcy subproblem, it is thus natural to resort to domain decomposition methods, which allows one to solve the coupled system sequentially with a low computational cost.…”

Section: Introductionmentioning

confidence: 99%

A Robin-type domain decomposition method for a novel mixed-type DG method for the coupled Stokes-Darcy problem

Zhao¹

2021

Preprint

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In this paper, we first propose and analyze a novel mixed-type DG method for the coupled Stokes-Darcy problem on simplicial meshes. The proposed formulation is locally conservative. A mixed-type DG method in conjunction with the stress-velocity formulation is employed for the Stokes equations, where the symmetry of stress is strongly imposed. The staggered DG method is exploited to discretize the Darcy equations. As such, the discrete formulation can be easily adapted to account for the Beavers-Joseph-Saffman interface conditions without introducing additional variables. Importantly, the continuity of normal velocity is satisfied exactly at the discrete level. A rigorous convergence analysis is performed for all the variables. Then we devise and analyze a domain decomposition method via the use of Robin-type interface boundary conditions, which allows us to solve the Stokes subproblem and the Darcy subproblem sequentially with low computational costs. The convergence of the proposed iterative method is analyzed rigorously. In particular, the proposed iterative method also works for very small viscosity coefficient. Finally, several numerical experiments are carried out to demonstrate the capabilities and accuracy of the novel mixed-type scheme, and the convergence of the domain decomposition method.

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An analysis on the penalty and Nitsche's methods for the Stokes–Darcy system with a curved interface

Cited by 10 publications

References 42 publications

An extended finite element method for coupled Darcy–Stokes problems

An extended finite element method for coupled Darcy–Stokes problems

Generalized Penalty Method for a Class of Variational-hemivariational Inequality

A Robin-type domain decomposition method for a novel mixed-type DG method for the coupled Stokes-Darcy problem

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