Upscaling of a class of nonlinear parabolic equations for the flow transport in heterogeneous porous media

Chen, Zhiming; Deng, Weibing; Huang, Ye

doi:10.4310/cms.2005.v3.n4.a2

Cited by 26 publications

(26 citation statements)

References 31 publications

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“…Notice that such an inverse assumption is often assumed for the analysis of FEM for non-linear problems [34,28,36,25,17]. In our analysis it is used only in the proof of an L 2 estimate (see Lemma 4.2) and for the uniqueness of the numerical solution (Sect.…”

Section: Resultsmentioning

confidence: 99%

Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems

Abdulle

Vilmart

2013

Math. Comp.

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Abstract. An analysis of the finite element heterogeneous multiscale method for a class of quasilinear elliptic homogenization problems of nonmonotone type is proposed. We obtain optimal convergence results for dimension d ≤ 3. Our results, which also take into account the microscale discretization, are valid for both simplicial and quadrilateral finite elements. Optimal a-priori error estimates are obtained for the H 1 and L 2 norms, error bounds similar as for linear elliptic problems are derived for the resonance error. Uniqueness of a numerical solution is proved. Moreover, the Newton method used to compute the solution is shown to converge. Numerical experiments confirm the theoretical convergence rates and illustrate the behavior of the numerical method.

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Section: Resultsmentioning

confidence: 99%

Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems

Abdulle

Vilmart

2013

Math. Comp.

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“…[1,9,14]. Various iterative methods for solving nonlinear equations have been proposed and studied in the past, e.g., [6,7,11,25,34,37].…”

Section: Introductionmentioning

confidence: 99%

Robust Multiscale Iterative Solvers for Nonlinear Flows in Highly Heterogeneous Media

Efendiev¹,

Galvis²,

Kang³

et al. 2012

NMTMA

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In this paper, we study robust iterative solvers for finite element systems resulting in approximation of steady-state Richards' equation in porous media with highly heterogeneous conductivity fields. It is known that in such cases the contrast, ratio between the highest and lowest values of the conductivity, can adversely affect the performance of the preconditioners and, consequently, a design of robust preconditioners is important for many practical applications. The proposed iterative solvers consist of two kinds of iterations, outer and inner iterations. Outer iterations are designed to handle nonlinearities by linearizing the equation around the previous solution state. As a result of the linearization, a large-scale linear system needs to be solved. This linear system is solved iteratively (called inner iterations), and since it can have large variations in the coefficients, a robust preconditioner is needed. First, we show that under some assumptions the number of outer iterations is independent of the contrast. Second, based on the recently developed iterative methods, we construct a class of preconditioners that yields convergence rate that is independent of the contrast. Thus, the proposed iterative solvers are optimal with respect to the large variation in the physical parameters. Since the same preconditioner can be reused in every outer iteration, this provides an additional computational savings in the overall solution process. Numerical tests are presented to confirm the theoretical results.

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“…We consider an exponential stationary model for the infiltration of a fluid in unsaturated porous media. We choose here an L-shape computational domain (similar test problems have been considered in [41,3] for regular domains). The multiscale tensor is defined as the tensor in the offline stage.…”

Section: 2mentioning

confidence: 99%

Adaptive reduced basis finite element heterogeneous multiscale method

Abdulle

Bai

2013

Computer Methods in Applied Mechanics and Engineering

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An adaptive reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) is proposed for elliptic problems with multiple scales. The multiscale method is based on the RB-FE-HMM introduced in [A. Abdulle and Y. Bai, J. Comput. Phys, 2012, in press]. It couples a macroscopic solver with effective data recovered from the solution of micro problems solved on sampling domains. Unlike classical numerical homogenization methods, the micro problems are computed in a finite dimensional space spanned by a small number of accurately computed representative micro solutions (the reduced basis) obtained by a greedy algorithm in an offline stage. In this paper we present a residual-based a posteriori error analysis in the energy norm as well as an a posteriori error analysis in quantities of interest. For both type of adaptive strategies, rigorous a posteriori error estimates are derived and corresponding error estimators are proposed. In contrast to the adaptive finite element heterogeneous multiscale method (FE-HMM), there is no need to adapt the micro mesh simultaneously to the macroscopic mesh refinement. Up to an offline preliminary stage, the RB-FE-HMM has the same computational complexity as a standard adaptive FEM for the effective problem. Two and three dimensional numerical experiments confirm the efficiency of the RB-FE-HMM and illustrate the improvements compared to the adaptive FE-HMM.

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Upscaling of a class of nonlinear parabolic equations for the flow transport in heterogeneous porous media

Cited by 26 publications

References 31 publications

Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems

Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems

Robust Multiscale Iterative Solvers for Nonlinear Flows in Highly Heterogeneous Media

Adaptive reduced basis finite element heterogeneous multiscale method

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