High-Order Multiscale Finite Element Method for Elliptic Problems

Hesthaven, Jan S.; Zhang, Shun; Zhu, Xueyu

doi:10.1137/120898024

Cited by 21 publications

(30 citation statements)

References 16 publications

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“…In this paper, we will use local versions of oscillating test functions to build multiscale finite element elements, following the ideas of [4] and [14].…”

Section: General Homogenization Resultsmentioning

confidence: 99%

“…Multiscale finite element methods based on local oscillating test functions. In this section, we define MsFEM in the spirit of [4,14]. We first solve to obtain oscillating test functions on the local elements or on larger patches if oversampling technique is used.…”

Section: General Homogenization Resultsmentioning

confidence: 99%

“…We first solve to obtain oscillating test functions on the local elements or on larger patches if oversampling technique is used. The multiscale finite element then is defined by a composite rule [4] or by a more direct approach [14]. Oversampling and Petrov-Galerkin formulations can be used to improve the accuracy of the methods.…”

Section: General Homogenization Resultsmentioning

confidence: 99%

“…Originally, MsFEM was proposed for linear finite elements but generalized in [4] to enable the use of high-order elements by local oscillating test functions (or harmonic coordinates). In [14], we proposed an alternative formulation of high-order MsFEM using related ideas, albeit introducing a more natural formulation.…”

mentioning

confidence: 99%

“…In this paper, we focus on the development of reduced basis multiscale finite element method (RB-MsFEM) for MsFEM based on local oscillating test functions( [4,14]). The central element of the reduced basis method is the parametrization of the problems to be solved.…”

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confidence: 99%

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Reduced Basis Multiscale Finite Element Methods for Elliptic Problems

Hesthaven¹,

Zhang²,

Zhu³

2015

Multiscale Model. Simul.

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Abstract.In this paper, we propose reduced basis multiscale finite element methods (RB-MsFEM) for elliptic problems with highly oscillating coefficients. The method is based on multiscale finite element methods with local test functions that encode the oscillatory behavior ([4, 14]). For uniform rectangular meshes, the local oscillating test functions are represented by a reduced basis method, parameterizing the center of the elements. For triangular elements, we introduce a slightly different approach. By exploring over-sampling of the oscillating test functions, initially introduced to recover a better approximations of the global harmonic coordinate map, we first build the reduced basis on uniform rectangular elements containing the original triangular elements and then restrict the oscillating test function to the triangular elements. These techniques are also generalized to the case where the coefficients dependent on additional independent parameters. The analysis of the proposed methods is supported by various numerical results, obtained on regular and unstructured grids.Key words. Multiscale finite element methods, reduced basis methods.AMS subject classifications. TBD 1. Introduction. The development of efficient and accurate numerical methods for solving problems with highly oscillating coefficients is an area of research that is increasingly active. This is not only driven by applications in subsurface flows or the modeling of novel materials, but also by the prohibitive cost of solving such problems using a straightforward approach where all scales are adequately resolved.To address and, ultimately overcome, the computational cost of resolving the finest scale, multiscale finite element methods (MsFEM) have been developed in [15,16,10,17,9]. In this approach, accuracy is achieved by locally solving a fine scale problem. These solutions are subsequently used to build the multiscale finite element basis, encoding the local fine structure, to capture the fine scale information of the leading order differential operator. Originally, MsFEM was proposed for linear finite elements but generalized in [4] to enable the use of high-order elements by local oscillating test functions (or harmonic coordinates). In [14], we proposed an alternative formulation of high-order MsFEM using related ideas, albeit introducing a more natural formulation.The local oscillating test functions must be solved for each element of the coarse mesh and each of these local problems must be resolve to fully capture the local fine scale. Hence, these local problems has many degrees of freedom (DOFs) and a fast solver of these local problems is central to the efficiency of MsFEM. To address this, we consider the reduced basis method (RBM) as an ideal technique that provides an efficient representation of the solution to parametric problems in manyquery and real-time scenarios, see introductions in [23,24, 25]. This idea has been pursued by a few authors recently in the context of multi-scale problems, as in [7], where reduced basis met...

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“…In this paper, we will use local versions of oscillating test functions to build multiscale finite element elements, following the ideas of [4] and [14].…”

Section: General Homogenization Resultsmentioning

confidence: 99%

Section: General Homogenization Resultsmentioning

confidence: 99%

Section: General Homogenization Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Reduced Basis Multiscale Finite Element Methods for Elliptic Problems

Hesthaven¹,

Zhang²,

Zhu³

2015

Multiscale Model. Simul.

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The localized reduced basis multiscale method for two‐phase flows in porous media

Kaulmann¹,

Flemisch

Haasdonk³

et al. 2014

Numerical Meth Engineering

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In this work, we propose a novel model order reduction approach for twophase flow in porous media by introducing a formulation in which the mobility, which realizes the coupling between phase saturations and phase pressures, is regarded as a parameter to the pressure equation. Using this formulation, we introduce the Localized Reduced Basis Multiscale method to obtain a low-dimensional surrogate of the high-dimensional pressure equation. By applying ideas from model order reduction for parametrized partial differential equations, we are able to split the computational effort for solving the pressure equation into a costly offline step that is performed only once and an inexpensive online step that is carried out in every time step of the two-phase flow simulation, which is thereby largely accelerated. Usage of elements from numerical multiscale methods allows us to displace the computational intensity between the offline and online step to reach an ideal runtime at acceptable error increase for the two-phase flow simulation.

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Multiscale seamless‐domain method for solving nonlinear problems using statistical estimation methodology

Suzuki

2017

Numerical Meth Engineering

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This article presents a nonlinear solver combining regression analysis and a multiscale simulation scheme. First, the proposed method repeats microscopic analysis of a local simulation domain, which is extracted from the entire global domain, to statistically estimate the relation(s) between the value of a dependent variable at a point and values at surrounding points. The relation is called regression function. Subsequent global analysis reveals the behavior of the global domain with only coarse-grained points using the regression function quickly at low computational cost, which can be accomplished using a multiscale numerical solver, called the seamless-domain method. The objective of the study is to solve a nonlinear problem accurately and at low cost by combining the 2 techniques. We present an example problem of a nonlinear steadystate heat conduction analysis of a heterogeneous material. The proposed model using fewer than 1000 points generates a solution with precision similar to that of a standard finite-element solution using hundreds of thousands of nodes. To investigate the relationship between the accuracy and computational time, we apply the seamless-domain method under varying conditions such as the number of iterations of the prior analysis for statistical data learning.KEYWORDS elliptic, multiscale, nonlinear solvers, partial differential equations, regression analysis | INTRODUCTIONPrevious work 1-4 presented a multiscale numerical solver called the seamless-domain method (SDM). The SDM model is meshless and represented by only a small number of coarse-grained points (CPs). The method constructs a structure as a "seamless" analytical domain whose distributions of a dependent variable and its gradient are almost continuous Nomenclature: Symbol, Explanation; # −1 , inverse of #; # T , transpose of #; Ω G ⊂ R d , domain for global analysis; Ω L ⊂ R d , domain for local analysis; Ω R ⊂ Ω L , region of influence; Γ G , boundary of the global domain; Γ L , boundary of the local domain; Γ R , boundary of the region of influence; a ∈ R 1 × m , weighting coefficient matrix; d ∈ {1, … , 3}, dimension of domain; f i , i-th response-surface function that determines the dependent-variable value at a coarse-grained point (CP) from the values at surrounding CPs (u R m ð Þ ); f L i , i-th response-surface function that determines u L m ð Þ from u R m ð Þ ; m, number of CPs in a region of influence; m l , number of iterations of the prior local analysis for constructing each response surface; m r , number of response surfaces; n, number of CPs in a global domain; N ∈ R 1 × m , interpolating function matrix for u R m ð Þ at temperature u ref ; R, set of all real numbers; u(x) ∈ R, dependent variable at point x; u G n ð Þ ∈R n , dependent variable of CPs in a global domain (Ω G ); u L m ð Þ ∈R m , dependent variable for all CPs near a local domain 0 s boundary (Γ L ); u R m ð Þ ∈R m , dependent variable for all CPs near a boundary of a region of influence (Γ R ); u ref i , reference temperature for CP i; x ∈ R d ,...

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High-Order Multiscale Finite Element Method for Elliptic Problems

Cited by 21 publications

References 16 publications

Reduced Basis Multiscale Finite Element Methods for Elliptic Problems

Reduced Basis Multiscale Finite Element Methods for Elliptic Problems

The localized reduced basis multiscale method for two‐phase flows in porous media

Multiscale seamless‐domain method for solving nonlinear problems using statistical estimation methodology

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