Open Access
Volume 55, Number 4, July-August 2021
Page(s) 1375 - 1404
Published online 07 July 2021
  1. A. Abdulle and P. Henning, Localized orthogonal decomposition method for the wave equation with a continuum of scales. Math. Comput. 86 (2017) 549–587. [Google Scholar]
  2. G. Avalos and I. Lasiecka, Optimal blowup rates for the minimal energy null control of the strongly damped abstract wave equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. 2 (2003) 601–616. [Google Scholar]
  3. J. Azevedo, C. Cuevas and H. Soto, Qualitative theory for strongly damped wave equations. Math. Methods Appl. Sci. 40 (2017) 08. [Google Scholar]
  4. I. Babuska and R. Lipton, Optimal local approximation spaces for generalized finite element methods with application to multiscale problems. SIAM J. Multiscale Model. Simul. 9 (2010) 373–406. [Google Scholar]
  5. I. Babuska and J.E. Osborn, Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20 (1983) 510–536. [CrossRef] [MathSciNet] [Google Scholar]
  6. E. Bonetti, E. Rocca, R. Scala and G. Schimperna, On the strongly damped wave equation with constraint. Commun. Part. Differ. Equ. 42 (2017) 1042–1064. [Google Scholar]
  7. A. Carvalho and J. Cholewa, Local well posedness for strongly damped wave equations with critical nonlinearities. Bull. Aust. Math. Soc. 66 (2002) 443–463. [Google Scholar]
  8. C. Cuevas, C. Lizama and H. Soto, Asymptotic periodicity for strongly damped wave equations. In: Vol. 2013 of Abstract and Applied Analysis. Hindawi (2013). [CrossRef] [Google Scholar]
  9. N. Dal Santo, S. Deparis, A. Manzoni and A. Quarteroni, Multi space reduced basis preconditioners for large-scale parametrized PDEs. SIAM J. Sci. Comput. 40 (2018) A954–A983. [Google Scholar]
  10. M. Drohmann, B. Haasdonk and M. Ohlberger, Adaptive reduced basis methods for nonlinear convection-diffusion equations. In: Vol. 4 of Finite Volumes for Complex Applications VI. Problems & Perspectives. Volume 1, 2, Springer Proc. Springer, Heidelberg (2011) 369–377. [Google Scholar]
  11. C. Engwer, P. Henning, A. Målqvist and D. Peterseim, Efficient implementation of the localized orthogonal decomposition method. Comput. Methods Appl. Mech. Eng. 350 (2019) 123–153. [Google Scholar]
  12. F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006) 185–207. [Google Scholar]
  13. P.J. Graber and J.L. Shomberg, Attractors for strongly damped wave equations with nonlinear hyperbolic dynamic boundary conditions. Nonlinearity 29 (2016) 1171. [Google Scholar]
  14. B. Haasdonk, M. Ohlberger and G. Rozza, A reduced basis method for evolution schemes with parameter-dependent explicit operators. Electron. Trans. Numer. Anal. 32 (2008) 145–161. [Google Scholar]
  15. P. Henning and A. Målqvist, Localized orthogonal decomposition techniques for boundary value problems. SIAM J. Sci. Comput. 36 (2014) A1609–A1634. [CrossRef] [Google Scholar]
  16. P. Henning, A. Målqvist and D. Peterseim, A localized orthogonal decomposition method for semi-linear elliptic problems. ESAIM: M2AN 48 (2014) 1331–1349. [CrossRef] [EDP Sciences] [Google Scholar]
  17. T.J. Hughes, G.R. Feijóo, L. Mazzei and J.-B. Quincy, The variational multiscale method – a paradigm for computational mechanics. Comput. Methods Appl. Mech. Eng. 166 (1998) 3–24. [CrossRef] [MathSciNet] [Google Scholar]
  18. R. Ikehata, Decay estimates of solutions for the wave equations with strong damping terms in unbounded domains. Math. Methods Appl. Sci. 24 (2001) 659–670. [Google Scholar]
  19. V. Kalantarov and S. Zelik, A note on a strongly damped wave equation with fast growing nonlinearities. J. Math. Phys. 56 (2015). [Google Scholar]
  20. P. Kelly, Solid Mechanics Part I: An Introduction to Solid Mechanics. University of Auckland (2019). [Google Scholar]
  21. A. Khanmamedov, Strongly damped wave equation with exponential nonlinearities. J. Math. Anal. Appl. 419 (2014) 663–687. [Google Scholar]
  22. M. Larson and A. Målqvist, Adaptive variational multiscale methods based on a posteriori error estimation: energy norm estimates for elliptic problems. Comput. Methods Appl. Mech. Eng. 196 (2007) 2313–2324. [Google Scholar]
  23. S. Larsson, V. Thomée and L.B. Wahlbin, Finite-element methods for a strongly damped wave equation. IMA J. Numer. Anal. 11 (1991) 115–142. [Google Scholar]
  24. Y. Lin, V. Thomée and L. Wahlbin, Ritz-Volterra projections to finite element spaces and applications to integro-differential and related equations. Technical report (Cornell University. Mathematical Sciences Institute). Mathematical Sciences Institute, Cornell University (1989). [Google Scholar]
  25. A. Målqvist and A. Persson, Multiscale techniques for parabolic equations. Numer. Math. 138 (2018) 191–217. [CrossRef] [PubMed] [Google Scholar]
  26. A. Målqvist and D. Peterseim, Generalized finite element methods for quadratic eigenvalue problems. ESAIM: M2AN 51 (2017) 147–163. [EDP Sciences] [Google Scholar]
  27. A. Målqvist and D. Peterseim, Numerical Homogenization beyond Periodicity and Scale Separation. To appear in SIAM Spotlight (2020). [Google Scholar]
  28. A. Målqvist and D. Peterseim, Computation of eigenvalues by numerical upscaling. Numer. Math. 130 (2012) 337–361. [Google Scholar]
  29. A. Målqvist and D. Peterseim, Localization of elliptic multiscale problems. Math. Comp. 83 (2014) 2583–2603. [Google Scholar]
  30. P. Massatt, Limiting behavior of strongly damped nonlinear wave equations. J. Differ. Equ. 48 (1983) 334–349. [Google Scholar]
  31. D. Peterseim, Variational multiscale stabilization and the exponential decay of fine-scale correctors. In: Vol. 114 of Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lect. Notes Comput. Sci. Eng. Springer, [Cham] (2016) 341–367. [Google Scholar]
  32. A. Quarteroni, A. Manzoni and F. Negri, Reduced Basis Methods for Partial Differential Equations: An Introduction. Springer (2016). [Google Scholar]
  33. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd edition. Springer Series in Computational Mathematics. Springer-Verlag, Berlin (2006). [Google Scholar]
  34. G.F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation. Can. J. Math. 32 (1980) 631–643. [Google Scholar]

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