The Mathematical Theory of Finite Element Methods

Brenner, Susanne C.; Scott, L. Ridgway

doi:10.1007/978-1-4757-3658-8

Cited by 2,781 publications

(3,254 citation statements)

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“…For Case 2B, because both U α,0 and v are periodic, then [17][18][19], then the discrete problem is to find…”

Section: (B) Continuous Problemmentioning

confidence: 99%

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A dual weighted residual method applied to complex periodic gratings

Lord

Mulholland²

2013

Proc. R. Soc. A.

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An extension of the dual weighted residual (DWR) method to the analysis of electromagnetic waves in a periodic diffraction grating is presented. Using the α,0-quasi-periodic transformation, an upper bound for the a posteriori error estimate is derived. This is then used to solve adaptively the associated Helmholtz problem. The goal is to achieve an acceptable accuracy in the computed diffraction efficiency while keeping the computational mesh relatively coarse. Numerical results are presented to illustrate the advantage of using DWR over the global a posteriori error estimate approach. The application of the method in biomimetic, to address the complex diffraction geometry of the Morpho butterfly wing is also discussed.

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“…For Case 2B, because both U α,0 and v are periodic, then [17][18][19], then the discrete problem is to find…”

Section: (B) Continuous Problemmentioning

confidence: 99%

“…Let X α be a finite-element subspace of order p of H 1 # (F) and let ζ h be any regular partition of X α [17,18,22,23]. Denote by h the maximum mesh size of the triangular elements in this partition.…”

Section: A Posteriori Error Estimationmentioning

confidence: 99%

A dual weighted residual method applied to complex periodic gratings

Lord

Mulholland²

2013

Proc. R. Soc. A.

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show abstract

“…[8,15,34]. Remarkably, for p = 1, the gain and loss terms converge at the same rate as the approximation itself, while for p = 2, the convergence rate of the gain and loss terms exceeds that of the approximation.…”

Section: Convergence Of the Dg Finite-element Approximationmentioning

confidence: 99%

A discontinuous Galerkin finite-element method for a 1D prototype of the Boltzmann equation

Hoitinga

Brummelen

2011

Journal of Computational Physics

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“…It follows from property (ii) of g and standard properties of discrete harmonic functions [2,6,23] that…”

Section: Two-level Additive Schwarz Preconditionermentioning

confidence: 99%

“…Since the derivation of (20) depends crucially on the discrete Sobolev inequality [2,6,23], v * is intimately related to piecewise linear functions on an interval with special property with respect to the Sobolev norm of order 1 2 . Let I = (0, 1).…”

Section: Bramble-pasciak-schatz Preconditionermentioning

confidence: 99%