Overlapping Schwarz and Spectral Element Methods for Linear Elasticity and Elastic Waves

Pavarino, Luca F.; Zampieri, E.

doi:10.1007/s10915-005-9047-7

Cited by 11 publications

(7 citation statements)

References 21 publications

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“…These tests show that the above good convergence properties also hold for heterogeneous problems with discontinuous coefficients across subdomain boundaries. Therefore, our isogeometric Schwarz preconditioners retain the good convergence properties of overlapping Schwarz solvers for standard Galerkin discretizations, for both h-finite elements (see, e.g., the books [37,38]) and hp or spectral elements (see, e.g., [32,33,25]). Our 2-level additive algorithm and analysis for IGA can be easily extended to 2-level multiplicative and hybrid versions, as in the finite and spectral element cases.…”

mentioning

confidence: 85%

Overlapping Schwarz Methods for Isogeometric Analysis

Veiga¹,

Cho²,

Pavarino³

et al. 2012

SIAM J. Numer. Anal.

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Abstract. We construct and analyze an overlapping Schwarz preconditioner for elliptic problems discretized with isogeometric analysis. The preconditioner is based on partitioning the domain of the problem into overlapping subdomains, solving local isogeometric problems on these subdomains, and solving an additional coarse isogeometric problem associated with the subdomain mesh. We develop an h-analysis of the preconditioner, showing in particular that the resulting algorithm is scalable and its convergence rate depends linearly on the ratio between subdomain and "overlap sizes" for fixed polynomial degree p and regularity k of the basis functions. Numerical results in two-and three-dimensional tests show the good convergence properties of the preconditioner with respect to the isogeometric discretization parameters h, p, k, number of subdomains N , overlap size, and also jumps in the coefficients of the elliptic operator.Key words. domain decomposition methods, overlapping Schwarz, scalable preconditioners, isogeometric analysis, finite elements, NURBS AMS subject classifications. 65N55, 65N30, 65F10 DOI. 10.1137/110833476 1. Introduction. Isogeometric analysis (IGA) based on NURBS (nonuniform rational B-splines) was introduced by Hughes, Cottrell, and Bazilevs in [27] as an innovative numerical methodology for the analysis of PDE problems, allowing for an exact description of CAD-type geometries. NURBS are a standard in the computeraided design (CAD) community mainly because such spline functions allow excellent representations of free-form surfaces, and there are very efficient algorithms for their evaluation, refinement, and derefinement. In the isogeometric framework, the NURBS basis functions representing the CAD geometry are also used as the basis for the discrete solution space of PDEs, following an isoparametric paradigm. In addition to exact representation of CAD geometries, another advantage of using NURBS basis functions is the higher control on the regularity of the discrete space. For instance, spaces of global C k regularity are easily built, thus allowing for fewer degrees of freedom, better performance in case of vibrations, easier approximation of higher order problems, and other advantages. IGA methodologies have been summarized in the recent book [18] and studied in, e.g., [2,4,9,10,19,23,28,29,11,5,8]. IGA methods are having a growing impact on fields as diverse as fluid dynamics [6,7,40,15,26], structural mechanics [3,1,12,20,30,39], and electromagnetics [17,16].The discrete systems produced by isogeometric methods are better conditioned than the systems produced by standard finite elements or finite differences, but their conditioning can still degenerates rapidly for decreasing mesh size h, increasing poly-

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mentioning

confidence: 85%

Overlapping Schwarz Methods for Isogeometric Analysis

Veiga¹,

Cho²,

Pavarino³

et al. 2012

SIAM J. Numer. Anal.

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“…Meshes are generated with Cubit from Sandia National Laboratory and partitioned with Parmetis [32,39]. We validate the structure part of the solver based on an analytic problem in [40] and the fluid problem part from [36]. The validation of the fluid-structure problem is discussed below.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Scalable parallel methods for monolithic coupling in fluid–structure interaction with application to blood flow modeling

Barker

Cai

2010

Journal of Computational Physics

111

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“…For the classical GLL SEM case with only one element per subdomain, several works are available in the literature, both for the scalar and vector cases, see e.g. [12,13,18,15]. Afterwards the more general case, where Ω is decomposed into subdomains Ω i 's which are in turn partitioned into spectral elements, has been studied for instance in [17] for both triangular SEM based on Fekete nodes and quadrilateral SEM based on GLL nodes.…”

Section: Overlapping Schwarz Preconditionersmentioning

confidence: 99%

“…Overlapping Schwarz methods for GLL spectral elements have been initially constructed for scalar elliptic problems (see e.g. [12,9]) and then successfully applied to important application areas such as Navier-Stokes problems [13,14], linear elasticity and elastic waves [15], and nontensorial and unstructured spectral elements [16][17][18]. The main goal of this paper is to show numerically that overlapping Schwarz methods can be successfully extended to GLL spectral elements employing Gordon-Hall transfinite mapping techniques, and in particular retain their good convergence properties such as scalability and optimality also for heterogeneous problems with discontinuity jumps in the elliptic coefficients.…”

Section: Introductionmentioning

confidence: 99%

Overlapping Schwarz preconditioners for spectral element methods in nonstandard domains and heterogeneous media

Ghezzi¹,

Pavarino

Zampieri

2010

Journal of Computational and Applied Mathematics

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a b s t r a c tOverlapping Schwarz preconditioners are constructed and numerically studied for GaussLobatto-Legendre (GLL) spectral element discretizations of heterogeneous elliptic problems on nonstandard domains defined by Gordon-Hall transfinite mappings. The results of several test problems in the plane show that the proposed preconditioners retain the good convergence properties of overlapping Schwarz preconditioners for standard affine GLL spectral elements, i.e. their convergence rate is independent of the number of subdomains, of the spectral degree in the case of generous overlap and of the discontinuity jumps in the coefficients of the elliptic operator, while in the case of small overlap, the convergence rate depends on the inverse of the overlap size.

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Overlapping Schwarz and Spectral Element Methods for Linear Elasticity and Elastic Waves

Cited by 11 publications

References 21 publications

Overlapping Schwarz Methods for Isogeometric Analysis

Overlapping Schwarz Methods for Isogeometric Analysis

Scalable parallel methods for monolithic coupling in fluid–structure interaction with application to blood flow modeling

Overlapping Schwarz preconditioners for spectral element methods in nonstandard domains and heterogeneous media

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