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Babuška, Ivo; Aziz, Abdul

doi:10.1016/b978-0-12-068650-6.50006-x

Cited by 126 publications

(26 citation statements)

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“…Our proof uses the scaling properties of the tetrahedralizations and the following important approximation result. [48][49][50][51] To state this result, let us recall the definition of the "degree m Lagrange interpolant. "…”

Section: Proofs Of Approximation Resultsmentioning

confidence: 99%

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Analysis of periodic Schrödinger operators: Regularity and approximation of eigenfunctions

Hunsicker

Nistor

Sofo

2008

Journal of Mathematical Physics

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Analysis of periodic Schrödinger operators: Regularity and approximation of eigenfunctionsLet V be a real valued potential that is smooth everywhere on R 3 , except at a periodic, discrete set S of points, where it has singularities of the Coulomb-type Z / r. We assume that the potential V is periodic with period lattice L. We study the spectrum of the Schrödinger operator H =−⌬ + V acting on the space of Bloch waves with arbitrary, but fixed, wavevector k. Let T ª R 3 / L. Let u be an eigenfunction of H with eigenvalue and let ⑀ Ͼ 0 be arbitrarily small. We show that the classical regularity of the eigenfunction u is u H 5/2−⑀ ͑T͒ in the usual Sobolev spaces, and u K 3/2−⑀ m ͑T \ S͒ in the weighted Sobolev spaces. The regularity index m can be as large as desired, which is crucial for numerical methods. For any choice of the Bloch wavevector k, we also show that H has compact resolvent and hence a complete eigenfunction expansion. The case of the hydrogen atom suggests that our regularity results are optimal. We present two applications to the numerical approximation of eigenvalues: using wave functions and using piecewise polynomials.

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Section: Proofs Of Approximation Resultsmentioning

confidence: 99%

“…[48][49][50][51] Theorem V.1: Let T be a tetrahedralization of a polyhedral domain P ʚ T with the property that all tetrahedra comprising T have angles Ն␣ and edges Յh. Then there exists an absolute constant C͑␣ , m͒ such that, for any u H m+1 ͑P͒,…”

Section: A Lagrange Interpolants and Approximationmentioning

confidence: 99%

Analysis of periodic Schrödinger operators: Regularity and approximation of eigenfunctions

Hunsicker

Nistor

Sofo

2008

Journal of Mathematical Physics

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“…For p.>O, x bounded away from x,, <jJ(x, p.) is known to have a representationc 5 J ( 9) Assume now that u(r, cp) is of the special form:…”

Section: Ll Behavior Ofmentioning

confidence: 99%

Properties of Transport Solution for Neutron Density in Slab Geometry in Piecewise Polynomial Approximation

Pitkäranta¹

1976

Journal of Nuclear Science and Technology

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The order of accuracy of finite-element type approximations, when applied to partial differential equations, depends generally on the smoothness of the exact solution, measured by means of Sobolev norms. The properties of the one-speed neutron density in multi-layer slab geometry are considered from this point of view. It is shown in particular that the neutron density admits finite Sobolev-integrals of order less than unity only. The integrability results are applied to derive asymptotic error estimates for piecewise polynomial variational approximations. The inclusion of proper singular functions in a polynomial trial space is shown to speed up the convergence.

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“…The next step is to let (7) is standard (see [1], [7]). The space S in this case is contained in -7-the space of all piecevise constant functions.…”

Section: The Discrete Kelvin Principlementioning

confidence: 99%