The Experts below are selected from a list of 2316 Experts worldwide ranked by ideXlab platform
Béatrice Rivière - One of the best experts on this subject based on the ideXlab platform.
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a high order hybridizable discontinuous galerkin method for incompressible Miscible Displacement in heterogeneous media
Results in Applied Mathematics, 2020Co-Authors: Maurice S Fabien, Matthew G Knepley, Béatrice RivièreAbstract:Abstract An hybridizable discontinuous Galerkin method of arbitrary high order is formulated to solve the Miscible Displacement problem in porous media. The spatial discretization is combined with a sequential algorithm that decouples the flow and the transport equations. Hybridization produces a linear system for the globally coupled degrees of freedom, that is smaller in size compared to the system resulting from the interior penalty discontinuous Galerkin methods. We study the impact of increasing the polynomial order on the accuracy of the solution. Numerical experiments show that the method converges optimally and that it is robust for highly heterogeneous porous media in two and three dimensions.
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a high order hybridizable discontinuous galerkin method for incompressible Miscible Displacement in heterogeneous media
arXiv: Computational Engineering Finance and Science, 2018Co-Authors: Maurice S Fabien, Matthew G Knepley, Béatrice RivièreAbstract:We present a new method for approximating solutions to the incompressible Miscible Displacement problem in porous media. At the discrete level, the coupled nonlinear system has been split into two linear systems that are solved sequentially. The method is based on a hybridizable discontinuous Galerkin method for the Darcy flow, which produces a mass--conservative flux approximation, and a hybridizable discontinuous Galerkin method for the transport equation. The resulting method is high order accurate. Due to the implicit treatment of the system of partial differential equations, we observe computationally that no slope limiters are needed. Numerical experiments are provided that show that the method converges optimally and is robust for highly heterogeneous porous media in 2D and 3D.
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Numerical solutions of the incompressible Miscible Displacement equations in heterogeneous media
Computer Methods in Applied Mechanics and Engineering, 2015Co-Authors: Jizhou Li, Béatrice RivièreAbstract:Abstract This paper presents a numerical method based on mixed finite element, discontinuous Galerkin methods in space and high order Runge–Kutta method in time for solving the Miscible Displacement problem. No slope limiters are needed. The proposed method exhibits high order of convergence in space and time when comparing with analytical solutions. The simulation shows robustness of the method for heterogeneous media with highly varying permeabilities.
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convergence of a high order method in time and space for the Miscible Displacement equations
Mathematical Modelling and Numerical Analysis, 2015Co-Authors: Jizhou Li, Béatrice Rivière, Noel J WalkingtonAbstract:A numerical method is formulated and analyzed for solving the Miscible Displacement problem under low regularity assumptions. The scheme employs discontinuous Galerkin time stepping with mixed and interior penalty discontinuous Galerkin finite elements in space. The numerical approx- imations of the pressure, velocity, and concentration converge to the weak solution as the mesh size and time step tend to zero. To pass to the limit a compactness theorem is developed which generalizes the Aubin−Lions theorem to accommodate discontinuous functions both in space and in time.
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convergence of a discontinuous galerkin method for the Miscible Displacement equation under low regularity
SIAM Journal on Numerical Analysis, 2011Co-Authors: Béatrice Rivière, Noel J WalkingtonAbstract:Discontinuous Galerkin time discretizations are combined with the mixed finite element and continuous finite element methods to solve the Miscible Displacement problem. Stable schemes of arbitrary order in space and time are obtained. Under low regularity assumptions on the data, convergence of the scheme is proved by using compactness results for functions that may be discontinuous in time.
Yanping Chen - One of the best experts on this subject based on the ideXlab platform.
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an efficient two grid method for Miscible Displacement problem approximated by mixed finite element methods
Computers & Mathematics With Applications, 2019Co-Authors: Yanping Chen, Yunqing Huang, Shang Liu, Jie ZhouAbstract:Abstract In the paper, we present an efficient two grid method for the Miscible Displacement problem which discretized by mixed finite element methods for the pressure equation and concentration equation at the same time, and then analyzed the error estimate of the two-gird algorithm. At last, the numerical experiment presented confirmed the theoretical results. Compared with the standard mixed finite element methods, this two-grid scheme based on the mixed methods can keep the same convergence order and cost much less work.
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a two grid method for incompressible Miscible Displacement problems by mixed finite element and eulerian lagrangian localized adjoint methods
Journal of Mathematical Analysis and Applications, 2018Co-Authors: Yang Wang, Yanping ChenAbstract:Abstract In this paper, we present a scheme for solving two-dimensional Miscible Displacement problems using Eulerian–Lagrangian localized adjoint methods and mixed finite element methods. Since only the velocity and not the pressure appears explicitly in the concentration equation, an Eulerian–Lagrangian localized adjoint method is used to solve the concentration equation and a mixed finite element method is used for the pressure equation. To linearize and decouple the mixed-method equations, we use a two-grid algorithm based on the Newton iteration method for this fully discrete problems. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid using Newton iteration once. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O ( h 1 / 2 ) in this paper.
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a priori error estimates of a combined mixed finite element and local discontinuous galerkin method for an incompressible Miscible Displacement problem
Applied Mathematics and Computation, 2018Co-Authors: Yanping Chen, Jiming Yang, Yunqing HuangAbstract:Abstract A numerical approximation for a kind of incompressible Miscible Displacement problems in high dimension in porous media is studied. Mixed finite element method is applied to the flow equation, and the transport one is solved by the local discontinuous Galerkin method (LDG). Based on interpolation projection properties and the induction hypothesis, a priori hp error estimates are obtained. Numerical results are presented, which verify the theoretical results.
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two grid methods for Miscible Displacement problem by galerkin methods and mixed finite element methods
International Journal of Computer Mathematics, 2018Co-Authors: Shang Liu, Yanping Chen, Yunqing Huang, Jie ZhouAbstract:The Miscible Displacement problem of one incompressible fluid is modelled by a nonlinear coupled system of two partial differential equations in porous media. One equation is elliptic form for the ...
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two grid method for compressible Miscible Displacement problem by cfem mfem
Journal of Computational and Applied Mathematics, 2018Co-Authors: Jiaoyan Zeng, Yanping ChenAbstract:Abstract This paper is concerned about the error analysis of two-grid method for compressible Miscible Displacement in porous medium. A characteristics finite element method (CFEM) is presented for the concentration equation to handle the convection part, and standard mixed finite element method (MFEM) is used for the pressure equation. Moreover, we linearize the equations based on the Newton iteration method, then, two-grid method is considered in this full discrete scheme problem. We prove the L p error estimates for the pressure, Darcy velocity, concentration variables in the two-grid method. It is shown that coarse space can be extremely coarse and we achieve asymptotically optimal approximation. Finally, numerical experiment indicates that two-grid method is a very effective method for solving Miscible Displacement problem.
Jiming Yang - One of the best experts on this subject based on the ideXlab platform.
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a priori error estimates of a combined mixed finite element and local discontinuous galerkin method for an incompressible Miscible Displacement problem
Applied Mathematics and Computation, 2018Co-Authors: Yanping Chen, Jiming Yang, Yunqing HuangAbstract:Abstract A numerical approximation for a kind of incompressible Miscible Displacement problems in high dimension in porous media is studied. Mixed finite element method is applied to the flow equation, and the transport one is solved by the local discontinuous Galerkin method (LDG). Based on interpolation projection properties and the induction hypothesis, a priori hp error estimates are obtained. Numerical results are presented, which verify the theoretical results.
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Superconvergence Analysis of a Full-Discrete Combined Mixed Finite Element and Discontinuous Galerkin Approximation for an Incompressible Miscible Displacement Problem
Acta Applicandae Mathematicae, 2015Co-Authors: Jiming Yang, Zhiguang XiongAbstract:For an incompressible Miscible Displacement problem, an effective time-stepping procedure is proposed. The mixed finite element method is applied to the flow equation with uniform meshes, and the transport equation is solved by a full discretized interior penalty discontinuous Galerkin method with regular partitions. Convolution of the Darcy velocity approximation with the Bramble-Schatz kernel function and averaging are applied in the evaluation of the coefficients in the Galerkin procedure for the concentration. A superconvergence estimate is presented.
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superconvergence of a full discrete combined mixed finite element and discontinuous galerkin method for a compressible Miscible Displacement problem
Numerical Methods for Partial Differential Equations, 2013Co-Authors: Jiming Yang, Yanping Chen, Zhiguang XiongAbstract:An efficient time-stepping procedure is investigated for a two-dimensional compressible Miscible Displacement problem in porous media in which the mixed finite element method with Raviart-Thomas space is applied to the flow equation, and the transport one is solved by the symmetric interior penalty discontinuous Galerkin approximation on Cartesian meshes. Based on the projection interpolations and the induction hypotheses, a superconvergence error estimate is obtained. During the analysis, an extension of the Darcy velocity along the Gauss line is also used in the evaluation of the coefficients in the Galerkin procedure for the concentration. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013
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Superconvergence of a combined mixed finite element and discontinuous Galerkin approximation for an incompressible Miscible Displacement problem
Applied Mathematical Modelling, 2012Co-Authors: Jiming Yang, Yanping ChenAbstract:Abstract A combined mixed finite element and discontinuous Galerkin approximation for an incompressible Miscible Displacement problem which includes molecular diffusion and dispersion in porous media is studied. That is to say, the mixed finite element method is applied to the flow equation, and the transport equation is solved by an interior penalty discontinuous Galerkin method. Convolution of the Darcy velocity approximation with the Bramble–Schatz kernel function and averaging are applied in the evaluation of the coefficients in the Galerkin procedure for the concentration. A superconvergence estimate is obtained. Numerical experimental results are presented to verify the theoretical analysis.
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a priori error analysis of a discontinuous galerkin approximation for a kind of compressible Miscible Displacement problems
Science China-mathematics, 2010Co-Authors: Jiming Yang, Yanping ChenAbstract:A kind of compressible Miscible Displacement problems which include molecular diffusion and dispersion in porous media are investigated. A symmetric interior penalty discontinuous Galerkin (SIPG) method is applied to the coupled system of flow and transport. Using the induction hypotheses instead of the cut-off operator and the interpolation projection properties, a priori hp error estimates are presented. The error bounds in L2(H1) norm for concentration and in L∞(L2) norm for velocity are optimal in h and suboptimal in p with a loss of power 1/2.
Mary F Wheeler - One of the best experts on this subject based on the ideXlab platform.
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adaptive enriched galerkin methods for Miscible Displacement problems with entropy residual stabilization
Journal of Computational Physics, 2017Co-Authors: Sanghyun Lee, Mary F WheelerAbstract:We present a novel approach to the simulation of Miscible Displacement by employing adaptive enriched Galerkin finite element methods (EG) coupled with entropy residual stabilization for transport. In particular, numerical simulations of viscous fingering instabilities in heterogeneous porous media and Hele-Shaw cells are illustrated. EG is formulated by enriching the conforming continuous Galerkin finite element method (CG) with piecewise constant functions. The method provides locally and globally conservative fluxes, which are crucial for coupled flow and transport problems. Moreover, EG has fewer degrees of freedom in comparison with discontinuous Galerkin (DG) and an efficient flow solver has been derived which allows for higher order schemes. Dynamic adaptive mesh refinement is applied in order to reduce computational costs for large-scale three dimensional applications. In addition, entropy residual based stabilization for high order EG transport systems prevents spurious oscillations. Numerical tests are presented to show the capabilities of EG applied to flow and transport. Enriched Galerkin approximations for coupled flow and transport system.Entropy residual stabilization for enriched Galerkin to stabilize higher order transport scheme.Dynamic mesh adaptivity for viscous fingering in two and three dimensions.
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a combined mixed finite element and discontinuous galerkin method for Miscible Displacement problem in porous media
2002Co-Authors: Shuyu Sun, Béatrice Rivière, Mary F WheelerAbstract:A combined method consisting of the mixed finite element method for flow and the discontinuous Galerkin method for transport is introduced for the coupled system of Miscible Displacement problem. A “cut-off” operator M is introduced in the discontinuous Galerkin formular in order to make the combined scheme converge. Optimal error estimates in L 2(H 1) for concentration and in L ∞(L 2) for velocity are derived.
Maurice S Fabien - One of the best experts on this subject based on the ideXlab platform.
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a high order hybridizable discontinuous galerkin method for incompressible Miscible Displacement in heterogeneous media
Results in Applied Mathematics, 2020Co-Authors: Maurice S Fabien, Matthew G Knepley, Béatrice RivièreAbstract:Abstract An hybridizable discontinuous Galerkin method of arbitrary high order is formulated to solve the Miscible Displacement problem in porous media. The spatial discretization is combined with a sequential algorithm that decouples the flow and the transport equations. Hybridization produces a linear system for the globally coupled degrees of freedom, that is smaller in size compared to the system resulting from the interior penalty discontinuous Galerkin methods. We study the impact of increasing the polynomial order on the accuracy of the solution. Numerical experiments show that the method converges optimally and that it is robust for highly heterogeneous porous media in two and three dimensions.
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a high order hybridizable discontinuous galerkin method for incompressible Miscible Displacement in heterogeneous media
arXiv: Computational Engineering Finance and Science, 2018Co-Authors: Maurice S Fabien, Matthew G Knepley, Béatrice RivièreAbstract:We present a new method for approximating solutions to the incompressible Miscible Displacement problem in porous media. At the discrete level, the coupled nonlinear system has been split into two linear systems that are solved sequentially. The method is based on a hybridizable discontinuous Galerkin method for the Darcy flow, which produces a mass--conservative flux approximation, and a hybridizable discontinuous Galerkin method for the transport equation. The resulting method is high order accurate. Due to the implicit treatment of the system of partial differential equations, we observe computationally that no slope limiters are needed. Numerical experiments are provided that show that the method converges optimally and is robust for highly heterogeneous porous media in 2D and 3D.