Discontinuous Galerkin finite element heterogeneous multiscale method for advection–diffusion problems with multiple scales

Abdulle, Assyr; Huber, Martin

doi:10.1007/s00211-013-0578-9

Cited by 15 publications

(15 citation statements)

References 31 publications

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“…This class of problems, normally refereed to as multiscale problem, are know to be very computational demanding and arise in many different areas of the engineering sciences, e.g., in porous media flow and composite materials. More precisely, we consider the following convection-diffusion equation: given any f ∈ L 2 (Ω) we seek u ∈ H 1 0 (Ω) = {v ∈ H 1 (Ω) | v| Γ = 0} such that −∇ · A∇u + b · ∇u = f in Ω, (1) is fulfilled in a weak sense, where Ω is the computational domain with boundary Γ. The multiscale coefficients A, b will be specified later.…”

Section: Introductionmentioning

confidence: 99%

An Adaptive Discontinuous Galerkin Multiscale Method for Elliptic Problems

Elfverson¹,

Georgoulis²,

Målqvist³

2013

Multiscale Model. Simul.

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We propose an discontinuous Galerkin local orthogonal decomposition multiscale method for convection-diffusion problems with rough, heterogeneous, and highly varying coefficients. The properties of the multiscale method and the discontinuous Galerkin method allows us to better cope with multiscale features as well as interior/boundary layers in the solution. In the proposed method the trail and test spaces are spanned by a corrected basis computed on localized patches of size O(H log(H −1 )), where H is the mesh size. We prove convergence rates independent of the variation in the coefficients and present numerical experiments which verify the analytical findings.

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Section: Introductionmentioning

confidence: 99%

An Adaptive Discontinuous Galerkin Multiscale Method for Elliptic Problems

Elfverson¹,

Georgoulis²,

Målqvist³

2013

Multiscale Model. Simul.

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“…We begin by discretizing the macro scale and for the moment we assume that the effective permeability a 0 (x) is known for every x ∈ Ω. Such macro problem discretization was already studied in the case of homogenization-based multiscale methods for elliptic problems [2,7].…”

Section: Macro Scalementioning

confidence: 99%

A Discontinuous Galerkin Reduced Basis Numerical Homogenization Method for Fluid Flow in Porous Media

Abdulle¹,

Budáč²

2017

SIAM J. Sci. Comput.

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We present a new conservative multiscale method for Stokes flow in heterogeneous porous media. The method couples a discontinuous Galerkin finite element method (DG-FEM) at the macroscopic scale for the solution of an effective Darcy equation with a Stokes solver at the pore scale to recover effective permeabilities at macroscopic quadrature points. To avoid costly computation of numerous Stokes problems throughout the macroscopic computational domain, the pore geometry is parametrized and a model order reduction algorithm is used to select representative microscopic geometries. Accurate Stokes solutions and related permeabilities are obtained for these representative geometries in an offline stage. In an online stage, the DG-FEM is computed with permeabilities recovered at the desired macroscopic quadrature points from the precomputed Stokes solutions. The multiscale method is shown to be mass conservative at the macro scale and the computational cost for the online stage is similar to the cost of solving a single scale Darcy problem. Numerical experiments for two and three dimensional problems illustrate the efficiency and the performance of the proposed method.

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“…We note that for some problems for which mass conservation is required, other macroscopic FE space should be used. We mention, for example, the discontinuous Galerkin FE-HMM proposed and analysed for elliptic problem in [35] and for advection-diffusion problem (with possible high Peclet number) in [36]. For each element K of the macropartition, we consider a quadrature formula (ω K j , x K j ) j=1,...,J with weights ω K j and nodes x K j fulfilling classical assumptions (see §4 and [37]).…”

Section: (A) Main Ingredientsmentioning

confidence: 99%

Numerical Homogenization

Abdulle¹

2015

Encyclopedia of Applied and Computational Mathematics

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One contribution of 13 to a Theme Issue 'Multi-scale systems in fluids and soft matter: approaches, numerics and applications' . A general framework to combine numerical homogenization and reduced-order modelling techniques for partial differential equations (PDEs) with multiple scales is described. Numerical homogenization methods are usually efficient to approximate the effective solution of PDEs with multiple scales. However, classical numerical homogenization techniques require the numerical solution of a large number of so-called microproblems to approximate the effective data at selected grid points of the computational domain. Such computations become particularly expensive for high-dimensional, time-dependent or nonlinear problems. In this paper, we explain how numerical homogenization method can benefit from reducedorder modelling techniques that allow one to identify offline and online computational procedures. The effective data are only computed accurately at a carefully selected number of grid points (offline stage) appropriately 'interpolated' in the online stage resulting in an online cost comparable to that of a single-scale solver. The methodology is presented for a class of PDEs with multiple scales, including elliptic, parabolic, wave and nonlinear problems. Numerical examples, including wave propagation in inhomogeneous media and solute transport in unsaturated porous media, illustrate the proposed method.

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Discontinuous Galerkin finite element heterogeneous multiscale method for advection–diffusion problems with multiple scales

Cited by 15 publications

References 31 publications

An Adaptive Discontinuous Galerkin Multiscale Method for Elliptic Problems

An Adaptive Discontinuous Galerkin Multiscale Method for Elliptic Problems

A Discontinuous Galerkin Reduced Basis Numerical Homogenization Method for Fluid Flow in Porous Media

Numerical Homogenization

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