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Mixed Generalized Multiscale Finite Element Methods and Applications
Eric T. Chung, Yalchin Efendiev, Chak Shing Lee
TL;DR
The paper addresses conservative flow simulation in heterogeneous media with reduced multiscale spaces. It develops a mixed GMsFEM using snapshot and offline velocity spaces, analyzes convergence, and studies oversampling. The reported results show accurate two-phase flow and transport with only a few basis functions per coarse edge when appropriate offline spaces are selected.
Problem
Flow in heterogeneous media requires reduced representations while retaining a mass-conservative velocity field for flow and transport calculations.
Method
The method couples GMsFEM snapshot and spectrally reduced offline velocity spaces with a mixed finite element formulation, and also investigates oversampling.
Results
Accurate solutions are obtained with few basis functions per coarse edge, including two-phase flow and transport results without updating multiscale basis functions in time.
Takeaways & Limitations
Appropriately selected offline spaces can provide good accuracy for heterogeneous flow and transport while using a reduced coarse-grid representation.
Takeaways & Limitations
The convergence analysis assumes that each interior coarse edge has an offline basis function with nonzero normal component, and the eigenvalue analysis assumes rapidly decaying eigenvalues.
Abstract
from arXiv · showhide
In this paper, we present a mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving flow in heterogeneous media. Our approach constructs multiscale basis functions following a GMsFEM framework and couples these basis functions using a mixed finite element method, which allows us to obtain a mass conservative velocity field. To construct multiscale basis functions for each coarse edge, we design a snapshot space that consists of fine-scale velocity fields supported in a union of two coarse regions that share the common interface. The snapshot vectors have zero Neumann boundary conditions on the outer boundaries and we prescribe their values on the common interface. We describe several spectral decompositions in the snapshot space motivated by the analysis. In the paper, we also study oversampling approaches that enhance the accuracy of mixed GMsFEM. A main idea of oversampling techniques is to introduce a small dimensional snapshot space. We present numerical results for two-phase flow and transport, without updating basis functions in time. Our numerical results show that one can achieve good accuracy with a few basis functions per coarse edge if one selects appropriate offline spaces.
1. Introduction.
The paper targets flow in heterogeneous, multiscale media where model reduction is needed, and develops a mixed GMsFEM that preserves mass conservation while reducing computational cost.
- Heterogeneous media contain multiple scales and high-contrast features, motivating coarse-grid model reduction through upscaling and multiscale methods.Examples include fractures and shale layers whose thickness is much smaller than the domain size.
- The mixed finite element framework is chosen to preserve mass conservation and provide fine-scale conservative velocities without post-processing.
- Multiscale velocity basis functions are constructed for each coarse edge and supported in the two coarse blocks sharing that edge.
- GMsFEM separates construction into snapshot and offline stages, using spectral decomposition to retain dominant modes with fewer basis functions.
- The method constructs velocity snapshots and offline spaces, uses piecewise constant pressure functions, and studies spectral problems motivated by analysis.
- Numerical experiments cover heterogeneous permeability, single- and two-phase flow, and transport without updating basis functions in time.Adding a few extra basis functions improves prediction accuracy.
2. Preliminaries.
The preliminaries formulate high-contrast flow in mixed form on nested coarse and fine grids, with piecewise constant pressure and edge-based multiscale velocity spaces.
- The model considers flow with heterogeneous high-contrast permeability κ and non-homogeneous Neumann boundary data.
- The mixed GMsFEM constructs velocity basis functions for v = −κ∇p while approximating pressure with piecewise constant functions.
- The coarse grid partitions the domain into coarse blocks, each refined by connected fine-grid blocks forming the fine grid.
- A coarse neighborhood ωi is the union of coarse-grid blocks adjacent to a coarse edge Ei.
- For each coarse edge, local multiscale velocity basis functions solve a problem with prescribed normal velocity on the edge and zero normal velocity on the neighborhood boundary.
- The mixed GMsFEM seeks a coarse velocity-pressure pair in VH × QH subject to the prescribed boundary flux conditions.
3. The construction of multiscale basis functions.
The method builds an extensive local snapshot space from fine-scale flux responses, then reduces it through local spectral problems to obtain a compact offline velocity space.
- The snapshot space contains local solutions associated with all fine-grid boundary conditions, while the offline space retains dominant modes selected by local spectral decomposition.The resulting reduced space is used for velocity approximation.
- 3.1. Snapshot space.: For each coarse edge, snapshot fields are computed in its coarse neighborhood with zero normal velocity on the outer boundary and prescribed fine-edge flux data on the coarse edge.The coarse edge is decomposed into fine-grid edges, and each snapshot activates one such edge.
- 3.1. Snapshot space.: The snapshot solutions are extended by zero outside their coarse neighborhoods and assembled into a global snapshot space.
- 3.2. Offline space.: The eigenvalue construction uses symmetric positive definite bilinear forms and produces offline basis functions spanning the velocity approximation space.
- 3.2. Offline space.: The offline space is formed by selecting the first li eigenfunctions from local spectral problems on each coarse neighborhood.
- 3.2. Offline space.: The first eigenvector can be chosen as the mixed MsFEM basis function, while large eigenvalues scale inversely with fine-mesh size.
- 3.3. Optimization viewpoint of the basis functions.: The optimization viewpoint interprets each selected eigenfunction as maximally separated from the span of previously selected eigenvectors.
4. Convergence of the mixed GMsFEM.
The convergence analysis separates projection and spectral approximation errors, yielding an estimate for the mixed GMsFEM velocity relative to the fine-grid solution.
- The convergence proof first projects the fine-grid velocity into the snapshot space and then estimates the difference between that projection and the GMsFEM solution.
- The proof uses the fact that the fine-grid projection belongs to both the fine-grid space and the snapshot space.
- The snapshot projection satisfies a local error estimate that is assembled over all coarse-grid blocks.
- The analysis assumes a nonzero normal component for at least one offline basis function associated with each interior coarse edge.
- The convergence theorem bounds the mixed GMsFEM velocity error using the selected eigenvalues and coarse-grid approximation terms.
- The two terms in the error estimate represent spectral-basis error and coarse-grid discretization error, respectively.
5. Oversampling approach.
The oversampling approach computes snapshots on larger domains and can reduce their dimension before spectral decomposition. This improves accuracy and supports low-dimensional approximations, especially for problems with scale separation.
- Oversampling computes local snapshots on larger domains and restricts their interior solutions to reduce boundary pollution.The method uses κ-harmonic extensions and local boundary data in oversampled regions.
- POD applied to the snapshot space produces a lower-dimensional approximation space.The reduced representation is formed from leading POD modes selected from the oversampled snapshot space.
- The oversampling space is assembled from local oversampling spaces and used directly as the multiscale velocity space.The local spaces are spanned by oversampled snapshots, and the resulting space replaces V_H in the numerical solve.
- The convergence analysis bounds the approximation through the first omitted eigenvalue and the POD truncation error.The reduced-space argument is repeated to obtain a convergence rate after establishing the POD approximation estimate.
- When the forcing is constant in the relevant oversampled and local regions, the parameter δ equals zero.For other cases, δ depends on the smoothness of κ and f; for homogenization problems, it can be small.
- Using the reduced snapshot space, the analysis repeats the non-oversampling argument and obtains a convergence rate.The estimate follows after controlling the POD representation error.
- A reduced oversampling snapshot space can yield a low-dimensional structure when the problem has scale separation.This advantage may not hold for the corresponding non-oversampling procedure.
- The reduced snapshot construction lowers the dimension used in the subsequent spectral decomposition.This is the principal computational advantage of applying POD before the spectral decomposition.
6. Numerical results.
The numerical experiments evaluate mixed GMsFEM for single-phase flow, oversampling, and two-phase flow and transport across heterogeneous permeability fields. Results show convergence with added basis functions, strong spectral-error decay, benefits from postprocessing and oversampling, and improved saturation accuracy with more bases per coarse edge.
- Single-phase flow: Adding basis functions improves single-phase convergence, while spectral errors reach machine precision and total errors approach a fixed error.This behavior is reported for permeability field κ1 and similarly for κ2.
- Single-phase flow: The first 10 eigenvalues decay sharply, and the first 11 eigenfunctions suffice for machine-precision spectral error, with error-eigenvalue correlation 0.99.The lack of further decay after the 11th eigenvalue indicates that additional basis functions are unnecessary in this setting.
- Single-phase flow: Postprocessed velocity is much more accurate than the velocity without postprocessing in the κ1, n = 200, N = 10 experiment.The comparison is reported using the relative velocity error Epf(v).
- Oversampling technique: Oversampling generally improves performance and yields a smaller snapshot space with similar accuracy, including for high-contrast permeability.For the high-contrast field κ1, oversampling reduces error while retaining a small-dimensional snapshot space.
- Two-phase flow and transport: For one two-phase test, three and five bases per edge reduce saturation errors to approximately 2%, while velocity-space dimensions rise to 1% and 1.4% of the fine-scale space.With one basis per edge, saturation error is about 4% to 9%, and the offline velocity space is about 0.5% of the fine-scale space.
- Two-phase flow and transport: In two-phase flow and transport, using five bases per edge reduces saturation error from 9.3% to 2.6% at t = 1000 and from 5.5% to 1.3% at t = 5000.Another permeability field shows larger reductions, from 18.8% to 3.6% at t = 1000 and from 20.7% to 5.3% at t = 5000.
7. Conclusions.
The mixed GMsFEM constructs mass-conservative flow solutions through systematic velocity-basis enrichment and applies them to single- and two-phase incompressible flow. Oversampling can reduce snapshot-space dimension and improve accuracy, while the reported saturation figures illustrate results for 1, 3, and 5 bases per coarse edge.
- The mixed GMsFEM constructs mass-conservative flow solutions and applies them to single- and two-phase incompressible flow.The paper also analyzes convergence and provides an alternative view of eigenvalue construction.
- Oversampling constructs snapshot vectors on larger regions, producing a much smaller snapshot space and potentially improving mixed GMsFEM accuracy.The paper notes that oversampling can be particularly helpful for problems with scale separation.
- FIG. 6.12 reports a saturation solution using 1 basis per coarse edge on an 11 × 11 coarse grid.
- FIG. 6.13 reports a saturation solution using 3 bases per coarse edge on an 11 × 11 coarse grid.
- FIG. 6.14 reports a saturation solution using 5 bases per coarse edge on an 11 × 11 coarse grid.