Resolvent estimates and numerical implementation for the homogenisation of one‐dimensional periodic mixed type problems

Franz, Sebastian; Waurick, Marcus

doi:10.1002/zamm.201700329

Cited by 11 publications

(11 citation statements)

References 18 publications

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“…Further reading on homogenisation problems can also be found in these references. The first step of combining homogenisation processes and evolutionary equations has been made in [135] and has had some profound developments for both quantitative and qualitative results; see [23,42,136,138]. Exercise 14.3 Prove the 'subsequence argument': Let X be a topological space and (x n ) n a sequence in X.…”

Section: Commentsmentioning

confidence: 99%

Continuous Dependence on the Coefficients II

Seifert

Trostorff

Waurick

2021

Evolutionary Equations

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This chapter is concerned with the study of problems of the form $$\displaystyle \left (\partial _{t,\nu }M_{n}(\partial _{t,\nu })+A\right )U_{n}=F $$ ∂ t , ν M n ( ∂ t , ν ) + A U n = F for a suitable sequence of material laws $$\left (M_{n}\right )_{n}$$ M n n when A ≠ 0. The aim of this chapter will be to provide the conditions required for convergence of the material law sequence to imply the existence of a limit material law M such that the limit U =limn→∞Un exists and satisfies $$\displaystyle \left (\partial _{t,\nu }M(\partial _{t,\nu })+A\right )U=F. $$ ∂ t , ν M ( ∂ t , ν ) + A U = F .

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Section: Commentsmentioning

confidence: 99%

Continuous Dependence on the Coefficients II

Seifert

Trostorff

Waurick

2021

Evolutionary Equations

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“…In [8], resolvent estimates of some kind are calculated in a way that depends on N via the Gelfand transform, before a numerical analysis is conducted. How this might somehow be converted to a resolvent estimate for the limit problem itself remains an interesting unanswered question.…”

Section: Possible Future Directionsmentioning

confidence: 99%

Optimal energy decay in a one-dimensional wave-heat system with infinite heat part

S.¹,

Seifert²

2020

Journal of Mathematical Analysis and Applications

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Harnessing the abstract power of the celebrated result due to Borichev and Tomilov (Math. Ann. 347:455-478, 2010, no. 2), we study the energy decay in a one-dimensional coupled wave-heat-wave system. We obtain a sharp estimate for the rate of energy decay of classical solutions by first proving a growth bound for the resolvent of the semigroup generator and then applying the asymptotic theory of C0-semigroups. The present article can be naturally thought of as an extension of a recent paper by Batty, Paunonen, and Seifert (J. Evol. Equ. 16:649-664, 2016) which studied a similar wave-heat system via the same theoretical framework.with all the functions being understood to have been extended by zero in ξ to the interval (−1, 1). If the solution is sufficiently regular, a routine 2010 Mathematics Subject Classification. 35M33, 35B40, 47D06 (34K30).

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“…Using Theorem 6.8 together with the assumptions (a) and (b) imposed on q, we infer from equation (15) by letting n → ∞…”

Section: A Div-curl Type Characterisationmentioning

confidence: 99%

“…This goes well beyond the available results in the literature. Equations having highly oscillatory change of type have also been analysed in [48,15,7]. In these references, however, the attention is restricted to 1 + 1-dimensional model examples.…”

Section: An Application To Maxwell's Equationsmentioning

confidence: 99%

Nonlocal H-convergence

Waurick

2018

Calc. Var.

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We introduce the concept of nonlocal H-convergence. For this we employ the theory of abstract closed complexes of operators in Hilbert spaces. We show uniqueness of the nonlocal H-limit as well as a corresponding compactness result. Moreover, we provide a characterisation of the introduced concept, which implies that local and nonlocal H-convergence coincide for multiplication operators. We provide applications to both nonlocal and nonperiodic fully time-dependent 3D Maxwell's equations on rough domains. The material law for Maxwell's equations may also rapidly oscillate between eddy current type approximations and their hyperbolic non-approximated counter parts. Applications to models in nonlocal response theory used in quantum theory and the description of meta-materials, to fourth order elliptic problems as well as to homogenisation problems on Riemannian manifolds are provided.

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Resolvent estimates and numerical implementation for the homogenisation of one‐dimensional periodic mixed type problems

Abstract: We study a homogenisation problem for problems of mixed type in the framework of evolutionary equations. The change of type is highly oscillatory. The numerical treatment is done by a discontinuous Galerkin method in time and a continuous Galerkin method in space.

Cited by 11 publications

References 18 publications

Continuous Dependence on the Coefficients II

Continuous Dependence on the Coefficients II

Optimal energy decay in a one-dimensional wave-heat system with infinite heat part

Nonlocal H-convergence

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