Optimization of numerical processes

Babuška, Ivo; Sobolev, S L

doi:10.21136/am.1965.102941

Cited by 25 publications

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“…In particular, u will be smooth on Ω if f is smooth on Ω. This well-posedness result is especially useful in practice for the numerical approximation of the solution u of Equation (1), see for example [6,12,16] among many possible references. In practice, however, it is rarely the case that Ω is smooth.…”

Section: Introductionmentioning

confidence: 99%

Weighted Sobolev spaces and regularity for polyhedral domains

Ammann¹,

Nistor²

2007

Computer Methods in Applied Mechanics and Engineering

166

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Dedicated to Ivo Babuška on the occasion of his 80th birthday.Abstract. We prove a regularity result for the Poisson problem −∆u = f , u| ∂P = g on a polyhedral domain P ⊂ R 3 using the Babuška-Kondratiev spaces K m a (P). These are weighted Sobolev spaces in which the weight is given by the distance to the set of edges [4,33]. In particular, we show that there is no loss of K m a -regularity for solutions of strongly elliptic systems with smooth coefficients. We also establish a "trace theorem" for the restriction to the boundary of the functions in K m a (P).

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Section: Introductionmentioning

confidence: 99%

Weighted Sobolev spaces and regularity for polyhedral domains

Ammann¹,

Nistor²

2007

Computer Methods in Applied Mechanics and Engineering

166

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“…If ((p") с D{Ä) and {Ä(p^ is a complete sequence in Я then ((/>") is a complete sequence in Я^. These statements are evident from (1,3).…”

Section: Finite Dimensional Approximationsmentioning

confidence: 84%

“…Let N be a compact set in Я. Then {cp^ с Я^2 is an almost optimal least square approximation of (1,1)…”

Section: Theorem 11 Under the Above Assumptions éF\a-^) = еп^'ла-щ A-% а-ч") =Hni-mentioning

confidence: 99%

“…As a simple application of the last theorem we have Theorem 16. Let A be in the form (4,2), (4,3) on the interval (0, l) and let the right-hand side f of (1,1) have all derivatives and let its compact support lie on (0,1) and ||/<"*i^||W||/'"'lk= = <")• Then |л-Y -ргл-yiu = О Л^^^^^У Proof. We put H = L^(0,1) and, according to the assumptions of/,/ belongs + 00 to H^ and, by Proposition 5,/is also an element of H\.…”

Section: ы'-1г'йmentioning

confidence: 99%

“…In this paper I want to describe how some of his ideas may be used in the examination of variational methods for boundary-value problems. Optimal and almost optimal formulae (irrespectively of universality) were examined partly in [1], [2]. Some of the following results were published in [8].…”

Section: Introductionmentioning

confidence: 99%

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Universal almost optimal variational methods

Milota¹

1971

Czech. Math. J.

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Adaptive Approximation

Lippold¹

1987

Z Angew Math Mech

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The adaptivity of transitions on the mesh sequence ℋ in some mesh refinement structure is defined as a selection strategy which is performed by means of a monotone local indicator on the set of cells in that structure. Adaptivity then turns out to be a local optimality property relative to some maximum or sum indicator on ℋ. – Adaptive transitions generate optimal mesh sequences. The notion of δ‐adaptivity is introduced to catch problems where only perturbed error indicators are available. δ‐adaptive transitions generate quasi‐optimal mesh sequences. – For the solution of one‐dimensional elliptic boundary value problems by linear finite elements δ‐adaptive transitions are defined by means of the local defects in the differential equation which are caused by the finite element solutions.

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Optimization of numerical processes

Cited by 25 publications

References 0 publications

Weighted Sobolev spaces and regularity for polyhedral domains

Weighted Sobolev spaces and regularity for polyhedral domains

Universal almost optimal variational methods

Adaptive Approximation

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