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Constraint Energy Minimizing Generalized Multiscale Finite Element Method

Eric T. Chung, Yalchin Efendiev, Wing Tat Leung

arXiv:1704.03193v1math.NAcs.NA

TL;DR

High-contrast multiscale problems call for coarse-grid methods whose accuracy does not depend on physical scales and contrast. The paper combines small-eigenvalue GMsFEM auxiliary spaces with constrained energy minimization on oversampled domains. It reports mesh-dependent convergence under suitable oversampling and a modified construction that reduces oversampling's contrast dependence.

  • Problem

    Coarse-grid performance for multiscale, high-contrast problems is difficult to make independent of physical scales and contrast, while mesh-dependent convergence with few basis functions remains a goal.

    Source: §1 · pp. 1–2

  • Method

    The method selects small-eigenvalue GMsFEM modes for an auxiliary space and constructs basis functions by constrained energy minimization in oversampled domains.

    Source: §1 · p. 2, §2 · p. 4

  • Results

    If oversampling is insufficient, errors are large and contrast-dependent; the modified algorithm removes that contrast dependence and makes oversampling sizes less sensitive to contrast.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The auxiliary space represents channels with a minimal number of basis functions, and the standard construction's convergence rate is H/Λ with oversampling size depending weakly on contrast.

    Source: §1 · p. 2

    The analysis assumes that the partition-of-unity functions are bilinear.

    Source: §4 · p. 14

Abstract

from arXiv · show

The main goal of this paper is to design multiscale basis functions within GMsFEM framework such that the convergence of method is independent of the contrast and linearly decreases with respect to mesh size if oversampling size is appropriately chosen. We would like to show a mesh-dependent convergence with a minimal number of basis functions. Our construction starts with an auxiliary multiscale space by solving local spectral problems. In auxiliary multiscale space, we select the basis functions that correspond to small (contrast-dependent) eigenvalues. These basis functions represent the channels (high-contrast features that connect the boundaries of the coarse block). Using the auxiliary space, we propose a constraint energy minimization to construct multiscale spaces. The minimization is performed in the oversampling domain, which is larger than the target coarse block. The constraints allow handling non-decaying components of the local minimizers. If the auxiliary space is correctly chosen, we show that the convergence rate is independent of the contrast (because the basis representing the channels are included in the auxiliary space) and is proportional to the coarse-mesh size (because the constrains handle non-decaying components of the local minimizers). The oversampling size weakly depends on the contrast as our analysis shows. The convergence theorem requires that channels are not aligned with the coarse edges, which hold in many applications, where the channels are oblique with respect to the coarse-mesh geometry. The numerical results confirm our theoretical results. In particular, we show that if the oversampling domain size is not sufficiently large, the errors are large. To remove the contrast-dependence of the oversampling size, we propose a modified construction for basis functions and present numerical results and the analysis.

1 Introduction

High-contrast multiscale problems need coarse-grid performance that does not depend on physical scales and contrast. The paper uses GMsFEM auxiliary spaces and constrained energy minimization to seek mesh-dependent convergence with few basis functions, and reports that oversampling size affects errors and a relaxed variant reduces its contrast dependence.

1 Introduction

  • High-contrast multiscale applications make coarse-grid selection difficult, motivating numerical performance independent of physical scales and contrast.Source: §1 · p. 1
  • GMsFEM identifies high-contrast channels through local spectral problems so they can receive separate basis functions without geometric interpretation.Source: §1 · p. 2
  • The target is mesh-dependent convergence with few basis functions, but high-contrast channel solutions may not decay locally.Source: §1 · p. 2
  • Small-eigenvalue auxiliary basis functions represent channels, while constrained energy minimization on oversampled domains handles non-decaying components and yields a convergence rate H/Λ.The auxiliary space is sized to the number of channels; the analysis finds oversampling size depends weakly (logarithmically) on contrast.Source: §1 · p. 2
  • Numerical tests use two heterogeneous permeability fields containing high-conductivity channels and inclusions, with auxiliary functions selected to include channelized features.Source: §1 · p. 2
  • If oversampling is insufficient, errors are large and contrast-dependent; the modified algorithm removes this contrast dependence and makes oversampling sizes less sensitive to contrast.Source: §1 · p. 3

2 Preliminaries

The preliminaries set the coarse/fine-grid framework and describe the two-stage basis construction. GMsFEM builds an offline space by spectral selection from snapshots, with convergence tied to the smallest omitted eigenvalue.

2 Preliminaries

  • A coarse mesh of size H is refined into fine-grid blocks of size h, assumed fine enough to resolve the solution.Source: §2 · p. 3
  • The method constructs auxiliary multiscale basis functions with GMsFEM, then performs constrained energy minimization in an oversampling domain.Source: §2 · p. 4

2.1 The basic concepts of Generalized Multiscale Finite Element Method

  • Snapshot spaces can accelerate convergence, impose problem-relevant restrictions, and reduce offline-space construction cost.Source: §2.1 · p. 4
  • The paper discusses harmonic snapshots formed as extensions of fine-grid boundary functions, with randomized boundary conditions noted as a way to reduce construction cost.Source: §2.1 · p. 4
  • GMsFEM selects dominant snapshot-space modes corresponding to the smallest eigenvalues; convergence is proportional to 1/Λ*, where Λ* is the smallest omitted eigenvalue.Source: §2.1 · p. 4

3 The construction of the multiscale basis functions

The construction selects local spectral modes for an auxiliary space, then builds basis functions through constrained energy minimization on oversampled domains. The resulting global space has an orthogonality property used in the convergence analysis.

3 The construction of the multiscale basis functions

  • Local spectral eigenfunctions with the smallest eigenvalues form each auxiliary space, and their global sum supplies modes used in the construction.Source: §3 · p. 5
  • The construction uses local bilinear forms and the auxiliary-space inner product to define projection and orthogonality.Source: §3 · pp. 5–6
  • Each coarse block is enlarged by m layers to define an oversampled domain for local multiscale basis construction.Source: §3 · p. 6
  • The local basis construction is motivated by a global basis construction, and the resulting multiscale finite element space is defined by its basis functions.Source: §3 · p. 7
  • The global multiscale space is orthogonal to the null space of the auxiliary-space projection, yielding a decomposition of V into the global space and its complement.Source: §3 · p. 7

4 Analysis

The analysis connects global-space approximation to localized multiscale solutions and gives a logarithmic oversampling condition for the error estimate. It also establishes decay of the local basis functions outside oversampled regions.

4 Analysis

  • Under sufficient regularity of f, the analysis says the preceding error result can be improved.The supplied passage does not specify the improved bound.Source: §4 · p. 8
  • Lemma 2 establishes a unique local constrained minimizer whose projection equals the prescribed auxiliary basis function.This supports the comparison between global and localized basis functions used in the error analysis.Source: §4 · pp. 9, 13
  • The multiscale basis functions are small outside the oversampled region, with exponential decay when D is not large.The decay rate depends on a factor involving Λ and D.Source: §4 · pp. 10, 14
  • Theorem 1 relates localized-solution error to global-basis error and states a logarithmic oversampling condition involving contrast and H.The analysis first estimates the global-space solution, then bounds the difference between global and localized bases.Source: §4 · pp. 7, 13

5 Numerical Result

Across two high-contrast test cases, the reported errors follow the predicted coarse-mesh convergence rates. Their performance also depends on oversampling: increasing its size improves errors, while fixed-size oversampling becomes less effective as contrast rises.

Test case 1

  • First-order energy-norm and second-order L2-norm convergence with respect to H are reported for test case 1.The first-order energy-norm rate matches the theoretical bound; the results are shown in Table 1.Source: §5 · p. 14, Table 1 · p. 15
  • For fixed contrast, increasing oversampling improves results; for fixed oversampling, increasing contrast deteriorates performance in test case 1.Table 2 compares oversampling-layer counts and contrast values.Source: Table 2 · p. 15, §5 · p. 15

Test case 2

  • Test case 2 achieves the theoretical convergence rate, and its error improves as oversampling layers increase.The setup uses four basis functions per coarse block because some blocks have four small eigenvalues; Table 3 reports results for varying H.Source: Table 3 · p. 15, §5 · pp. 15–16

6 Relaxed constraint energy minimizing generalized multiscale finite element method

The relaxed method reduces the oversampling layers needed to achieve good results across the tested contrasts, while its analysis gives a convergence-rate bound independent of D. Its numerical results also show that the auxiliary space must include enough eigenfunctions to represent the channels.

6 Relaxed constraint energy minimizing generalized multiscale finite element method

  • The relaxed construction replaces the constrained minimization with an unconstrained one and gives an equivalent variational formulation.The spaces used in the formulation are defined as before.Source: §6 · p. 16

6.1 Analysis

  • The analysis gives an improved convergence bound whose rate is independent of D.The theorem’s estimate is discussed as an improvement over Theorem 1.Source: §6.1 · pp. 19–20

6.2 Numerical Result

  • With enough auxiliary eigenfunctions, the relaxed method achieves the predicted coarse-mesh convergence and improves on the original method’s oversampling-layer count in test case 1.The error results are shown in Table 5.Source: Table 5 · p. 20, §6.2 · p. 20
  • The relaxed method needs fewer oversampling layers at a fixed contrast and handles larger contrasts with a fixed number of layers in both test cases.The comparisons are reported in Tables 6–8.Source: §6.2 · p. 20, Table 6 · p. 21, Table 7 · p. 21
  • For the second medium, four auxiliary eigenfunctions represent its four high-contrast channels; fewer perform poorly, while more do not necessarily improve results.Table 9 reports the basis-count comparison.Source: Table 9 · p. 21, §6.2 · p. 21

7 Conclusions

The paper proposes Constraint Energy Minimizing GMsFEM, using small-eigenvalue auxiliary modes and local constrained energy minimization in oversampled domains. It reports contrast-independent convergence when oversampling is appropriately selected, and a relaxed constraint that reduces oversampling sensitivity to contrast.

7 Conclusions

  • Constraint Energy Minimizing GMsFEM constructs oversampled multiscale bases by minimizing energy subject to orthogonality constraints from a small-eigenvalue auxiliary space.The auxiliary space captures subgrid information that cannot be localized.Source: §7 · p. 21
  • With an appropriately selected oversampling domain, the method’s convergence is independent of contrast, though the main theorem’s required oversampling size depends on contrast.Source: §7 · p. 21
  • Relaxing the constraint removes the contrast dependence of oversampling size, and the required minimum number of auxiliary basis functions for good accuracy does not depend on contrast.The numerical results are reported as confirming the theoretical findings.Source: §7 · p. 22
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