Effective spectral approximations of convection—diffusion equations

Pasquarelli, Franco; Quarteroni, Alfio

doi:10.1016/s0045-7825(94)80006-5

Cited by 14 publications

(10 citation statements)

References 7 publications

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“…A different approach, more germane to those introduced in Section 8.3.2 for finite elements, has been proposed by Pasquarelli and Quarteroni (1994).…”

Section: Stabilization Methods For Spectral Approximationmentioning

confidence: 99%

Numerical Approximation of Partial Differential Equations

Quarteroni

Valli

1994

Springer Series in Computational Mathematics

1,468

1,496

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This book deals with the numerical approximation of partial differential equations. Its scope is to provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is one of its main features. Many kinds of problems are addressed. A comprehensive theory of Galerkin method and its variants, as well as that of collocation methods, are developed for the spatial discretization. These theories are then specified to two numerical subspace realizations of remarkable interest: the finite element method and the spectral method

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“…A different approach, more germane to those introduced in Section 8.3.2 for finite elements, has been proposed by Pasquarelli and Quarteroni (1994).…”

Section: Stabilization Methods For Spectral Approximationmentioning

confidence: 99%

Numerical Approximation of Partial Differential Equations

Quarteroni

Valli

1994

Springer Series in Computational Mathematics

1,468

1,496

View full text Add to dashboard Cite

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“…Many stabilization strategies have been proposed for standard ÿnite elements and ÿnite di erences in the last 20 years (see the books by Morton [1] and Roos et al [2]). Fewer works are available for spectral methods and spectral elements: Galerkin Least Square (GaLS), Streamline Upwind Petrov-Galerkin (SUPG), Douglas-Wang (DW) stabilizations for spectral discretizations have been studied by Pasquarelli and Quarteroni [11], Gervasio [16] and Gervasio and Saleri [18]; bubble stabilizations have been studied by Canuto [12], Canuto and Puppo [13] and Canuto et al [14].…”

Section: Stabilized Spectral Elementsmentioning

confidence: 99%

“…The literature for stabilized spectral and hp discretizations is less developed. For spectral methods, we refer to the works of Pasquarelli and Quarteroni [11], Canuto [12], Canuto and Puppo [13], Canuto et al [14], Funaro [15]. For spectral element methods, we refer to Gervasio [16], Funaro [17], Gervasio and Saleri [18].…”

Section: Introductionmentioning

confidence: 99%

Overlapping Schwarz preconditioners for spectral element discretizations of convection‐diffusion problems

Pavarino

2001

Numerical Meth Engineering

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SUMMARYThe classical overlapping Schwarz algorithm is here extended to stabilized spectral element discretizations of convection-di usion problems. The algorithm solves iteratively the resulting non-symmetric system of linear equations by a preconditioned GMRES method. The preconditioner is built from local convection-di usion solvers on overlapping subdomains and from a coarse convection-di usion solver on a coarse mesh deÿned by the subdomain boundaries. Several numerical experiments on test problems in the plane indicate that this algorithm retains the fast convergence rate and optimal scalability properties of classical overlapping methods for di usion dominated problems. Fast convergence is also obtained for convection dominated problems without closed streamlines and with a moderate number of subdomains.

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“…The combination of VMS, shock capturing and FOS methods will be shown to be an encouraging direction to take for constructing high-order positive-definite spectral element methods. To assess the algorithm, steady and transient advection-diffusion problems are solved on one-and two-dimensional domains using spectral elements up to order 16. We compare the performance of the method using VMS against those obtained with previous classical spectral element schemes.…”

Section: Main Contribution Of This Papermentioning

confidence: 99%

“…In the regions where discontinuities occur, FOS subdivides a spectral element of order p into p 2 subcells and then uses 1st-order basis functions and integration rules on every subcell of the element. The algorithms are assessed with the solution of classical steady and transient 1D, 2D, and pseudo-3D problems using spectral elements up to order 16.…”

Section: Introductionmentioning

confidence: 99%