Superconvergence of a full‐discrete combined mixed finite element and discontinuous Galerkin method for a compressible miscible displacement problem

Yang, Jiming; Chen, Yanping; Zhang, Xiong

doi:10.1002/num.21777

Cited by 19 publications

(4 citation statements)

References 24 publications

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“…Some special numerical techniques were introduced to control jumps of numerical approximations as well as the nonlinearality of the convection term. Moreover, the superconvergence results based on the DG methods were also proved in [39,40]. Besides the above, there were also significant works discussing the discontinuous Galerkin methods for incompressible miscible displacements, see e.g.…”

Section: Introductionmentioning

confidence: 93%

Bound-Preserving Discontinuous Galerkin Method for Compressible Miscible Displacement in Porous Media

Guo¹,

Yang²

2017

SIAM J. Sci. Comput.

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In this paper, we develop bound-preserving discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacement problems. We consider the problem with two components and the (volumetric) concentration of the ith component of the fluid mixture, c i , should be between 0 and 1. However, c i does not satisfy the maximum principle. Therefore, the numerical techniques introduced in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied directly. The main idea is to apply the positivity-preserving techniques to both c 1 and c 2 , respectively and enforce c 1 + c 2 = 1 simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure dp/dt as a source in the concentration equation. Moreover, c ′ i s are not the conservative variables, as a result, the classical bound-preserving limiter in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied. Therefore, another limiter will be introduced. Numerical experiments will be given to demonstrate the accuracy in L ∞ -norm and good performance of the numerical technique.

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Section: Introductionmentioning

confidence: 93%

Bound-Preserving Discontinuous Galerkin Method for Compressible Miscible Displacement in Porous Media

Guo¹,

Yang²

2017

SIAM J. Sci. Comput.

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“…Discontinuous Galerkin (DG) methods belong to a class of non-conforming methods (see [1][2][3][4][5][6][7][8][9][10] and they solve the differential equations by piecewise polynomial functions over a finite element space without any requirement on inter-element continuity-however, continuity on inter-element boundaries together with boundary conditions is weakly enforced through the bilinear form. DG is very attractive for practical numerical simulations because of its physical and numerical properties.…”

Section: Introductionmentioning

confidence: 99%

A two‐grid discontinuous Galerkin method for an incompressible miscible displacement problem

Yang

Zhou

2023

Numerical Methods Partial

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An incompressible miscible displacement problem in porous media is studied. A two-grid algorithm based on an interior penalty discontinuous Galerkin method is proposed. With this algorithm, solving a nonlinear system on the discontinuous finite element space is reduced to solving a nonlinear problem on a coarse grid and a linear problem on a fine grid. The error estimate for the concentration in H 1 -norm and the error estimate for the velocity in L 2 -norm are obtained. The numerical experiments are provided to confirm our theoretical analysis.

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“…Thus these methods have become an important and powerful tool to deal with discontinuous problems, see [12–15]. Primal DGFE methods for miscible displacement problems with interior penalty were presented in [14–19]. However, some difficulties are encountered when the DGFE method is directly applied to the higher‐order differential equations.…”

Section: Introductionmentioning

confidence: 99%

A combined hybrid mixed element method for incompressible miscible displacement problem with local discontinuous Galerkin procedure

Zhang

Han

Guo

et al. 2020

Numerical Methods Partial

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In this article, we propose a combined hybrid discontinuous mixed finite element method for miscible displacement problem with local discontinuous Galerkin method. Here, to obtain more accurate approximation and deal with the discontinuous case, we use the hybrid mixed element method to approximate the pressure and velocity, and use the local discontinuous Galerkin finite element method for the concentration. Compared with other combined methods, this method can improve the efficiency of computation, deal with the discontinuous problem well and keep local mass balance. We study the convergence of this method and give the corresponding optimal error estimates in L ∞ (L 2) for velocity and concentration and the super convergence in L ∞ (H 1) for pressure. Finally, we also present some numerical examples to confirm our theoretical analysis.

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Superconvergence of a full‐discrete combined mixed finite element and discontinuous Galerkin method for a compressible miscible displacement problem

Cited by 19 publications

References 24 publications

Bound-Preserving Discontinuous Galerkin Method for Compressible Miscible Displacement in Porous Media

Bound-Preserving Discontinuous Galerkin Method for Compressible Miscible Displacement in Porous Media

A two‐grid discontinuous Galerkin method for an incompressible miscible displacement problem

A combined hybrid mixed element method for incompressible miscible displacement problem with local discontinuous Galerkin procedure

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