IX. Parabolic Equations

Lax, Peter D.; Milgram, A. N.

doi:10.1515/9781400882182-010

Cited by 106 publications

(76 citation statements)

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“…be the completion in || • ||fc of Co°°(i2), the C° functions with compact support in D. Let Ait), for each fixed t e R + , be an elliptic operator of order 2p with constant coefficients (see, for example, Friedman [4,Chapter 10]). By the results of Lax and Milgram [9], there exists a real number y(t) such that A(t) -y(t)IesJ and 3iAit)) = 3(A(t)-y(t)I) = 772p(fí) n 77°(Q) independent of r. If the coefficients of Ait) depend continuously on t, then y(-) can be taken to be a continuous function of t. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use…”

Section: T(s;a(a))t(t;a(r))f(x)= F (T(t; A(r))f)(x-z)g°(zs) Dzmentioning

confidence: 95%

Abstract evolution equations

Goldstein¹

1969

Trans. Amer. Math. Soc.

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Section: T(s;a(a))t(t;a(r))f(x)= F (T(t; A(r))f)(x-z)g°(zs) Dzmentioning

confidence: 95%

Abstract evolution equations

Goldstein¹

1969

Trans. Amer. Math. Soc.

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“…In the construction of Sobolev gradients for problems in nonlinear partial differential equations [6] a family of problems of a somewhat similar nature arises. Using an idea of Beurling and Deny [2], [3] we give a generalization of the LaxMilgram Theorem [5] which unifies a wide class of Sobolev gradient constructions.…”

Section: Introductionmentioning

confidence: 99%

Laplacians and Sobolev gradients

Neuberger

1998

Proc. Amer. Math. Soc.

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“…In terms of the Lax Milgram theorem [124] the conditions of boundness and Uellipticity of the bilinear form a directly imply the well-possedness of the problem in the sense of Hadamard, and therefore Problem 2.3.5 admits the unique solution. For more details see [92].…”

Section: Linear Elasticitymentioning

confidence: 99%

Variational formulations and functional approximation algorithms in stochastic plasticity of materials

Rosić¹

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A class of abstract stochastic variational inequalities of the second kind described by uncertain parameters is considered within the framework of infinitesimal and large displacement elastoplasticity theory. Particularly the focus is set on the rateindependent evolutionary problem with general hardening whose material characteristics are assumed to have positively-definite distributions. By exhibiting the structure of the evolutionary equations in a convex setting the mathematical formulation is carried over to the computationally more suitable mixed variational description for which the existence and uniqueness of the solution is studied. Time discretised as usual with backward Euler, the inequality is reduced to a minimisation problem for a convex functional on discrete tensor product subspaces whose unique minimiser is obtained via a stochastic closest point projection algorithm based on "white noise analysis". To this end a description in the language of non-dissipative and dissipative operators is used, both employing the stochastic Galerkin method in its fully intrusive or non-intrusive variant. The former method represents the direct, purely algebraic way of computing the response in each iteration of Newton-like methods. As the solution is given in a form of polynomial chaos expansion, i.e. an explicit functional relationship between the independent random variables, the subsequent evaluations of its functionals (the mean, variance, or probabilities of exceedence) are shown to be very cheap, but with limited accuracy. Due to this reason, the intrusive method is contrasted to the less efficient but more accurate non-intrusive variant which evaluates the residuum in each iteration via high-dimensional integration rules based on random or deterministic sampling -Monte Carlo and related techniques. In addition to these, the problem is also solved with the help of the stochastic collocation method via sparse grid techniques. Finally, the methods are validated on a series of test examples in plain strain conditions whose reference solution is computed via direct integration methods. 5 RezimeU okviru teorije malih i velikih plastičnih deformacija razmatrana je klasa apstraktnih stohastičkih varijacionih nejednakosti opisanih slučajnim promenljivim. Poseban fokus je stavljen na asocijativni evolucioni problem sa generalnim ojačanjem cije materijalne karakteristike imaju distribuciju odredenu zakonom maksimalne entropije. Proučavajući strukturu evolucionih jednačina uz pomoć konveksne teorije analizirani su uslovi za postojanje i jedinstvenost rešenja uz dodatnu matematičku reformulaciju problema u numerički prikladan mešoviti varijacioni opis. Dobijena nejednakost se nakon implicitne diskretizacije svodi na minimizaciju konveksnog funkcionala definisanog u tenzorskom prostoru. Rešenje tako postavljenog problema se može dobiti novouvedenom stohastičkom metodom projekcije najbliže tačke uz pomoć teorije analize belogšuma. Pomenuta metoda se sastoji od dva koraka: elastičnog i plastičnog,čiji su algoritmi ...

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IX. Parabolic Equations

Cited by 106 publications

References 0 publications

Abstract evolution equations

Abstract evolution equations

Laplacians and Sobolev gradients

Variational formulations and functional approximation algorithms in stochastic plasticity of materials

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