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Spectral/hp Element Methods for Computational Fluid Dynamics$
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George Karniadakis and Spencer Sherwin

Print publication date: 2005

Print ISBN-13: 9780198528692

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198528692.001.0001

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DIFFUSION EQUATION

DIFFUSION EQUATION

Chapter:
(p.251) 5 DIFFUSION EQUATION
Source:
Spectral/hp Element Methods for Computational Fluid Dynamics
Author(s):

George Em Karniadakis (Contributor Webpage)

Spencer J. Sherwin

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198528692.003.0005

This chapter considers the diffusion equation: an implicit in time discretization leads to the Helmholtz equation. Both the temporal discretization and eigenspectra of second-order operators that dictate time-step restrictions are discussed. Appropriate preconditioning techniques for inversion of the stiffness matrix, non-smooth solutions due to geometric singularities, and recent advances in three-dimensional domains are discussed. The exercises at the end of the chapter build on the exercises of Chapters 3 and 4 to implement a two-dimensional standard Galerkin hp solution to the Helmholtz problem.

Keywords:   Galerkin discretization, temporal discretization, eigenspectra, non-smooth domains, Helmholtz problem, spectral/hp element solver

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