Evaluation of Local Multiscale Approximation Spaces for Partition of Unity Methods

Schweitzer, Marc Alexander; Wu, Sa

doi:10.1007/978-3-319-51954-8_9

Cited by 2 publications

(4 citation statements)

References 18 publications

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“…This is consistent with the simulations given in [42] for a medium with separated holes. This is also consistent with the more recent numerical investigation carried out for scalar and elastic problems using the optimal local basis functions in [43]. The theoretical proof of contrast independent exponential convergence for MS-GFEM applied to heterogeneous elastic and conduction problems is given in a forthcoming publication.…”

Section: Introductionsupporting

confidence: 86%

“…For much larger problems with many patches greater savings from computational costs are anticipated by using coarser hat functions on the boundary to generate local approximation spaces. We finish by noting that in [43] related work has been recently carried out showing the contrast independence of the local spectral basis functions on their domains and that the global solutions converge exponentially in the L 2 norm. There it is found that the local n-width basis functions outperform all other contemporary choices of local trial fields constructed by oversampling.…”

Section: Discussionmentioning

confidence: 89%

“…Here we prescribe hat functions on the boundary ∂ω * i and compute their A-harmonic extensions into ω * i as in (43) and (44). Then the basis for the local approximation space is given by the restriction of these functions to ω i ⊂ ω * i .…”

Section: Span Of Hat Functions Used Directly As Local Approximation S...mentioning

confidence: 99%

“…Another strategy [14] provides a method based on rSVD for choosing boundary data and using their A-harmonic extensions as a sub optimal basis and demonstrating a nearly exponential decay in the expected value of the error. In a recent independent development the numerical investigations [43] show that local bases constructed by A-harmonically extending traces of harmonic polynomials deliver exponential decay of the error.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Multiscale-Spectral GFEM and optimal oversampling

Babuška

Lipton

Sinz

et al. 2020

Computer Methods in Applied Mechanics and Engineering

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In this work we address the Multiscale Spectral Generalized Finite Element Method (MS-GFEM) developed in [I. Babuška and R. Lipton, Multiscale Modeling and Simulation 9 (2011), pp. 373-406]. We outline the numerical implementation of this method and present simulations that demonstrate contrast independent exponential convergence of MS-GFEM solutions. We introduce strategies to reduce the computational cost of generating the optimal oversampled local approximating spaces used here. These strategies retain accuracy while reducing the computational work necessary to generate local bases. Motivated by oversampling we develop a nearly optimal local basis based on a partition of unity on the boundary and the associated A-harmonic extensions.

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Section: Introductionsupporting

confidence: 86%

Section: Discussionmentioning

confidence: 89%

Section: Span Of Hat Functions Used Directly As Local Approximation S...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multiscale-Spectral GFEM and optimal oversampling

Babuška

Lipton

Sinz

et al. 2020

Computer Methods in Applied Mechanics and Engineering

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Efficient modeling of internal cracks for Laplace problem by XFEM using Joukowski mapping

Nakasumi

Schweitzer

2019

Numerical Meth Engineering

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Summary In this study, the extended finite element method (XFEM) is applied to the two‐dimensional Laplace equation with an internal discontinuity. The real part of a complex velocity potential from potential flow theory is used to represent the enrichment function in this technique. The Joukowski mapping, which maps a circle to a line, is mainly used to obtain a solution around an airfoil in two‐dimensional potential flow; here, we extend that solution to model magnetic flux around an internal crack. The effectiveness of the proposed method is verified using numerical examples of single and multiple cracks. The L2 error norm is used to evaluate the accuracy of the proposed method in comparison with XFEM using previously proposed enrichment functions (Heaviside and analytical forms for a single crack tip). The proposed method gives better results than those of the existing XFEM in the case of a coarse mesh.

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Evaluation of Local Multiscale Approximation Spaces for Partition of Unity Methods

Cited by 2 publications

References 18 publications

Multiscale-Spectral GFEM and optimal oversampling

Multiscale-Spectral GFEM and optimal oversampling

Efficient modeling of internal cracks for Laplace problem by XFEM using Joukowski mapping

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