Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Scientific Computation)

By C. Canuto, M. Hussaini, Alfio Maria Quarteroni, Thomas A. Zang. 2007

1,339 citations recorded by the source corpus.

Spectral methods, particularly in their multidomain version, have become firmly established as a mainstream tool for scientific and engineering computation. This book provides an extensive and critical overview of the essential algorithmic and theoretical aspects of spectral methods for complex geometries, in addition to detailed discussions of spectral algorithms for fluid dynamics in simple and complex geometries. Modern strategies for constructing spectral approximations in complex domains, such as spectral elements, mortar elements, and discontinuous Galerkin methods, as well as patching collocation, are introduced, analyzed in their mathematical aspects, and demonstrated by means of numerous numerical examples. Efficient domain decomposition preconditioners (of both Schwarz and Schur type) that are amenable to parallel implementation are deeply investigated. Representative simulations from continuum mechanics are also shown

Subfields: Control and Systems Engineering, Mechanical Engineering

View Paper: https://www.semanticscholar.org/paper/d4c669202ad7565c71ebea6a66d7a409d4a71cb0

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Citations in the corpus

The phase field method for geometric moving interfaces and their numerical approximations (2019, neutral): “For problems defined on regular computational domains but not with periodic boundary conditions, other spectral implementations are possible such as Chebyshev spectral methods for Dirichlet boundary conditions, see discussions in the following books and recent surveys [55, 75, 215, 222, 297, 355, 46”
A localized reduced-order modeling approach for PDEs with bifurcating solutions (2018, methodology): “For the numerical results reported in § 3.2 and § 3.3, we consider a ﬁnite element method [5,17,40] and a spectral element method [6,7,22] to eﬀect spatial discretization.”
Algorithm 902: GPOPS, A MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method (2010, methodology): “, the problem is divided into a relatively small number of subintervals and global collocation is performed across each subinterval) [Canuto et al. 2007].”
Reduced Bézier element quadrature rules for quadratic and cubic splines in isogeometric analysis (2014, background): “We conclude that in addition to the Gauss-Lobatto rules whose stability has been established elsewhere (see for instance [45, 46, 96]), the center-edge rule and the face rule also lead to stable reduced Bézier element quadrature rules that need no further stabilizing mechanisms.”
Strong formulation finite element method for arbitrarily shaped laminated plates - Part I. Theoretical analysis (2014, methodology): “Together with FEM, the Spectral Element Method (SEM) represents another domain decomposition technique currently in use as in (Canuto et al. 2007 and Ostachowicz et al. 2011).”
The Finite Cell Method: A Review in the Context of Higher-Order Structural Analysis of CAD and Image-Based Geometric Models (2015, methodology): “Higher-order approximations are used in the p-version of the finite element method [175, 176] and spectral element methods [31,32,71,88].”