Convergence Analysis of Crank–Nicolson Galerkin–Galerkin FEMs for Miscible Displacement in Porous Media

Cai, Wentao; Wang, Jilu; Wang, Kai

doi:10.1007/s10915-020-01194-0

Cited by 10 publications

(5 citation statements)

References 33 publications

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“…Some slightly different schemes were investigated by several authors, for example, see [25,33]. In this paper, we only focus on the scheme (2.9)-(2.11), while analyses for some other schemes are similar.…”

Section: Galerkin-mixed Fem and The Main Resultsmentioning

confidence: 99%

“…Before proving our main theorem, we shall present several useful lemmas in this section. With the solution

{({}, (C^{n}, U^{n}, P^{n}))}_{n = 0}^{N}

to the time‐discrete system, the error functions can be split into

\{\begin{matrix} left \\ {(‖‖, C_{h}^{n} - c^{n})}_{L^{2}} \leq {(‖‖, C_{h}^{n} - C^{n})}_{L^{2}} + {(‖‖, C^{n} - c^{n})}_{L^{2}}, (3.6) \\ {(‖‖, U_{h}^{n} - u^{n})}_{L^{2}} \leq {(‖‖, U_{h}^{n} - U^{n})}_{L^{2}} + {(‖‖, U^{n} - u^{n})}_{L^{2}}, (3.7) \\ {(‖‖, P_{h}^{n} - p^{n})}_{L^{2}} \leq {(‖‖, P_{h}^{n} - P^{n})}_{L^{2}} + {(‖‖, P^{n} - p^{n})}_{L^{2}} . (3.8) \end{matrix}

The estimates for the second parts of the above splitting and the regularity of the solution of the time‐discrete system (3.1)–(3.5) were given in [25], Theorem 3.1 and [33], Theorem 3.2, respectively, for slightly different schemes. We present these results in the following lemma and the proof is omitted.Lemma Suppose that the initial ‐ boundary value problem (1.1)–(1.5) has a unique solution ( c , u , p ) which satisfies (2.4).…”

Section: Discussionmentioning

confidence: 99%

“…The estimates for the second parts of the above splitting and the regularity of the solution of the time-discrete system (3.1)-(3.5) were given in [25], Theorem 3.1 and [33], Theorem 3.2, respectively, for slightly different schemes. We present these results in the following lemma and the proof is omitted.…”

Section: An Elliptic Quasi-projectionmentioning

confidence: 99%

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Optimal error analysis of Crank–Nicolson lowest‐order Galerkin‐mixed finite element method for incompressible miscible flow in porous media

Gao

Sun

2020

Numerical Methods Partial

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Numerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest-order Galerkin-mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest order Raviart-Thomas mixed element pair is used for the Darcy velocity and pressure. The existing error estimate of the method in L 2-norm is in the order O(h p + h 2 c) in spatial direction, which however is not optimal and valid only under certain extra restrictions on both time step and spatial meshes, excluding the most commonly used mesh h = h p = h c. This paper focuses on new and optimal error estimates of a linearized Crank-Nicolson lowest-order Galerkin-mixed finite element method (FEM), where the second-order accuracy for the concentration in both time and spatial directions is established unconditionally. The key to our optimal error analysis is an elliptic quasi-projection. Moreover, we propose a simple one-step recovery technique to obtain a new numerical Darcy velocity and pressure of second-order accuracy. Numerical results for both two and three-dimensional models are provided to confirm our theoretical analysis.

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Section: Galerkin-mixed Fem and The Main Resultsmentioning

confidence: 99%

“…Before proving our main theorem, we shall present several useful lemmas in this section. With the solution

{({}, (C^{n}, U^{n}, P^{n}))}_{n = 0}^{N}

to the time‐discrete system, the error functions can be split into

\{\begin{matrix} left \\ {(‖‖, C_{h}^{n} - c^{n})}_{L^{2}} \leq {(‖‖, C_{h}^{n} - C^{n})}_{L^{2}} + {(‖‖, C^{n} - c^{n})}_{L^{2}}, (3.6) \\ {(‖‖, U_{h}^{n} - u^{n})}_{L^{2}} \leq {(‖‖, U_{h}^{n} - U^{n})}_{L^{2}} + {(‖‖, U^{n} - u^{n})}_{L^{2}}, (3.7) \\ {(‖‖, P_{h}^{n} - p^{n})}_{L^{2}} \leq {(‖‖, P_{h}^{n} - P^{n})}_{L^{2}} + {(‖‖, P^{n} - p^{n})}_{L^{2}} . (3.8) \end{matrix}

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Optimal error analysis of Crank–Nicolson lowest‐order Galerkin‐mixed finite element method for incompressible miscible flow in porous media

Gao

Sun

2020

Numerical Methods Partial

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“…In [39], an extra Lipschitz continuous reaction term was introduced for some applications and the lowest-order mixed FEM was used for both concentration equation and pressure equation. A Crank-Nicolson scheme with the coupled convection term (U n+1 h •∇C n+1 h , φ h ) was proposed in [10]. Moreover, a fully implicit scheme was studied in [24,42], where an extra inner iteration was required at each time step for solving a system of nonlinear equations.…”

Section: Schemes and Main Resultsmentioning

confidence: 99%

New analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media

Sun

2020

Preprint

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Analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media has been investigated extensively in the last several decades. Of particular interest in practical applications is the lowest-order Galerkin-mixed method, in which a linear Lagrange FE approximation is used for the concentration and the lowest-order Raviart-Thomas FE approximation is used for the velocity/pressure. The previous works only showed the first-order accuracy of the method in L 2 -norm in spatial direction, which however is not optimal and valid only under certain extra restrictions on both time step and spatial mesh. In this paper, we provide new and optimal L 2 -norm error estimates of Galerkin-mixed FEMs for all three components in a general case. In particular, for the lowest-order Galerkin-mixed FEM, we show unconditionally the second-order accuracy in L 2 -norm for the concentration. Numerical results for both two and three-dimensional models are presented to confirm our theoretical analysis. More important is that our approach can be extended to the analysis of mixed FEMs for many strongly coupled systems to obtain optimal error estimates for all components.

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“…We propose implicit and semi-implicit time stepping discretizations for each of the formulations (3.4), (3.6) and (3.9). Few second order accurate time-stepping schemes were proposed for Richards equation based on the Crank-Nicolson method [19,79,80] and BDF methods [7,26] in their fully implicit form requering Newton or Picard iterations. The Crank-Nicolson method is A-stable but lacks L-stability, and may lead to non-monotone solutions for larger time steps.…”

Section: Temporal Discretizationmentioning

confidence: 99%

Implicit and semi-implicit second-order time stepping methods for the Richards equation

Keita

Beljadid

Bourgault

2021

Advances in Water Resources

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This study concerns numerical methods for efficiently solving the Richards equation where different weak formulations and computational techniques are analyzed. The spatial discretizations are based on standard or mixed finite element methods. Different implicit and semi-implicit temporal discretization techniques of second-order accuracy are studied. To obtain a linear system for the semi-implicit schemes, we propose second-order techniques using extrapolation formulas and/or semi-implicit Taylor approximations for the temporal discretization of nonlinear terms. A numerical convergence study and a series of numerical tests are performed to analyze efficiency and robustness of the different schemes. The developed scheme, based on the proposed temporal extrapolation techniques and the mixed formulation involving the saturation and pressure head and using the standard linear Lagrange element, performs better than other schemes based on the saturation and the flux and using the Raviart-Thomas elements. The proposed semi-implicit scheme is a good alternative when implicit schemes meet convergence issues.

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Convergence Analysis of Crank–Nicolson Galerkin–Galerkin FEMs for Miscible Displacement in Porous Media

Cited by 10 publications

References 33 publications

Optimal error analysis of Crank–Nicolson lowest‐order Galerkin‐mixed finite element method for incompressible miscible flow in porous media

Optimal error analysis of Crank–Nicolson lowest‐order Galerkin‐mixed finite element method for incompressible miscible flow in porous media

New analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media

Implicit and semi-implicit second-order time stepping methods for the Richards equation

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