Finite element heterogeneous multiscale method for nonlinear monotone parabolic homogenization problems

Abdulle, Assyr; Huber, Martin

doi:10.1051/m2an/2016003

Cited by 15 publications

(28 citation statements)

References 34 publications

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“…The need for such auxiliary adjoint micro problems is known for purely diffusive linear and nonlinear micro problems with non-symmetric tensors, e.g., see [10,27] and [6]. Regularity assumptions.…”

Section: Fully Discrete Analysis Of Spatial Macro and Micro Errorsmentioning

confidence: 99%

Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows

Abdulle

Huber

2017

Numer. Math.

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We introduce a numerical homogenization method based on a discontinuous Galerkin finite element heterogeneous multiscale method (DG-HMM) to efficiently approximate the effective solution of parabolic advection-diffusion problems with rapidly varying coefficients, large Péclet number and compressible flows. To estimate the missing data of an effective model, numerical upscaling is performed which accurately captures the effects of microscopic solenoidal or gradient flow at a macroscopic scale such as enhancement or depletion of the effective diffusion. For compressible flow with periodic data, we derive sharp a priori error estimates for the macro and micro discretization errors which are robust in the advection dominated regime. Numerical tests confirm the error estimates for problems with periodic data and illustrate the applicability of our method for problems with non-periodic data.

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Section: Fully Discrete Analysis Of Spatial Macro and Micro Errorsmentioning

confidence: 99%

Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows

Abdulle

Huber

2017

Numer. Math.

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“…This follows from the condition (Q1) and the hypotheses (A 0−2 ). Hence Problems (36) and (38) have a unique solution (see [8] for details). Optimale convergence rates for the FE-HMM (in the H 1 and L 2 norms) for arbitrary simplicial elements for (33) have been obtained in [7].…”

Section: Numerical Homogenization and Numerical Integration For Nonlimentioning

confidence: 99%

The role of numerical integration in numerical homogenization

Abdulle

2015

ESAIM: Proc.

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Abstract. Finite elements methods (FEMs) with numerical integration play a central role in numerical homogenization methods for partial differential equations with multiple scales, as the effective data in a homogenization problem can only be recovered from a microscopic solver at a finite number of points in the computational domain. In a multiscale framework the convergence of a FEM with numerical integration applied to the effective (homogenized) problem guarantees that the so-called macroscopic solver is consistent and convergent. Convergence results for FEM with numerical integration are however scarce in the literature and need often to be derived as a first step to analyze a numerical homogenization method for a given problem. In this paper we review and explain the main ideas in deriving convergence results for FEM with numerical integration for linear and nonlinear elliptic problems and explain the role of these methods in numerical homogenization.Résumé. Les méthodes d'éléments finis avec intégration numérique par quadrature jouent un rôle central dans l'homogénéisation numérique deséquations aux dérivées partielles multi-échelles. En effet, les coefficients de l'équation homogénéisée ne peuventêtre déterminés que pour un nombre fini de points du domaine considéré. Dans le cadre des méthodes multi-échelles basées sur un schéma macroscopique avec intégration numérique en la variable macroscopique et coupléesà des schémas microscopiques autour des points de quadratures, la convergence d'une méthode d'élément fini avec intégration numérique pour le problème homogénéisé garantit que la méthode macroscopique est consistante et convergente. Comme les résultats de convergence pour les méthodes d'éléments finis avec intégration numérique ne sont connus que pour un nombre limité de problèmes, des nouveaux résultats de ce type sont alors un premier pas indispensable pourétablir la convergence d'une méthode d'homogénéisation numérique pour un problème donné. Dans ce papier, nous effectuons un survol des idées clés pour l'analyse des méthodes d'éléments finis avec intégration numérique pour des problèmes linéaires et non linéaires et expliquons le rôle de ces méthodes dans l'homogénéisation numérique.

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“…The aim of this paper is to analyze the convergence of a linearized version of a nonlinear homogenization method proposed in [6] to approximate the effective solution to the multiscale problem ∂ t u ε − div(A ε (x, ∇u ε )) = f which includes the class of problems (1) for A ε (x, ξ) = a ε (x, ξ)ξ. The method of [6] combines a nonlinear FE-HMM method (coupling macro and micro finite element methods) with the implicit Euler method in time.…”

Section: Introductionmentioning

confidence: 99%

“…The method of [6] combines a nonlinear FE-HMM method (coupling macro and micro finite element methods) with the implicit Euler method in time. Although the computational cost of the nonlinear method in [6] is independent of the small scale ε, its upscaling procedure however relies on nonlinear elliptic cell problems which is computationally costly for practical simulations. The linearized version presented in this paper permits to avoid Newton iterations, which considerably improves the computational efficiency of the method.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Linearized Numerical Homogenization Method for Nonlinear Monotone Parabolic Multiscale Problems

Abdulle¹,

Huber²,

Vilmart³

2015

Multiscale Model. Simul.

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We introduce and analyze an efficient numerical homogenization method for a class of nonlinear parabolic problems of monotone type in highly oscillatory media. The new scheme avoids costly Newton iterations and is linear at both the macroscopic and the microscopic scales. It can be interpreted as a linearized version of a standard nonlinear homogenization method. We prove the stability of the method and derive optimal a priori error estimates which are fully discrete in time and space. Numerical experiments confirm the error bounds and illustrate the efficiency of the method for various nonlinear problems.Keywords: monotone parabolic multiscale problem, linearized scheme, numerical homogenization method, fully discrete a priori error estimates.

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Finite element heterogeneous multiscale method for nonlinear monotone parabolic homogenization problems

Cited by 15 publications

References 34 publications

Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows

Numerical homogenization method for parabolic advection–diffusion multiscale problems with large compressible flows

The role of numerical integration in numerical homogenization

Linearized Numerical Homogenization Method for Nonlinear Monotone Parabolic Multiscale Problems

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