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May  2014, 13(3): 1017-1044. doi: 10.3934/cpaa.2014.13.1017

Homogenisation theory for Friedrichs systems

Krešimir Burazin 1, and  Marko Vrdoljak 2,

1. 

Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, 31 Osijek, Croatia
2. 

Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10 Zagreb, Croatia

Received  March 2013 Revised  September 2013 Published  December 2013

We develop a general homogenisation procedure for Friedrichs systems. Under reasonable assumptions, the concepts of $G$ and $H$-convergence are introduced. As Friedrichs systems can be used to represent various boundary or initial-boundary value problems for partial differential equations, some additional assumptions are needed for compactness results. These assumptions are particularly examined for the stationary diffusion equation, the heat equation and a model example of a first order equation leading to memory effects. In the first two cases, the equivalence with the original notion of $H$-convergence is proved.
Keywords: compensated compactness., symmetric positive systems, H-convergence, Homogenisation, G-convergence.
Mathematics Subject Classification: Primary: 35B27, 35F45, 35M32; Secondary: 47F0.
Citation: Krešimir Burazin, Marko Vrdoljak. Homogenisation theory for Friedrichs systems. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1017-1044. doi: 10.3934/cpaa.2014.13.1017
References:
[1]

Grégoire Allaire, Shape Optimization by the Homogenization Method, Springer-Verlag, 2002.  Google Scholar

[2]

Nenad Antonić and Krešimir Burazin, Graph spaces of first-order linear partial differential operators, Math. Communications, 14 (2009), 135-155.  Google Scholar

[3]

Nenad Antonić and Krešimir Burazin, Intrinsic boundary conditions for Friedrichs systems, Comm. Partial Diff. Eq., 35 (2010), 1690-1715.  Google Scholar

[4]

Nenad Antonić and Krešimir Burazin, Boundary operator from matrix field formulation of boundary conditions for Friedrichs systems, J. Diff. Eq., 250 (2011), 3630-3651,  Google Scholar

[5]

Nenad Antonić, Krešimir Burazin and Marko Vrdoljak, Second-order equations as Friedrichs systems, Nonlinear Analysis: RWA, 14 (2014), 290-305.  Google Scholar

[6]

Nenad Antonić, Krešimir Burazin and Marko Vrdoljak, Heat equation as a Friedrichs system, Journal of Mathematical Analysis and Applications, 404 (2013), 537-553.  Google Scholar

[7]

Nenad Antonić and Marko Vrdoljak, Parabolic H-convergence and small-amplitude homogenization, Applicable Analysis, 88 (2009), 1493-1508.  Google Scholar

[8]

Alain Bensoussan, Jacques-Lous Lions and Georgios Papanicolaou, Asymptotic Analysis in Periodic Structures, Northćolland, Amsterdam, 1978. Google Scholar

[9]

Krešimir Burazin, Contributions to the theory of Friedrichs' and Hyperbolic Systems, Ph.D thesis (in Croatian), University of Zagreb, 2008, http://www.mathos.hr/~kburazin/papers/teza.pdf. Google Scholar

[10]

Tan Bui-Thanh, Leszek Demkowicz and Omar Ghattas, A unified discontinuous Petrov-Galerkin method and its analysis for Friedrichs' systems, SIAM J. Numer. Anal., 51 (2013), 1933-1958.  Google Scholar

[11]

Andrea Dall'Aglio and Francois Murat, A corrector result for $H$-converging parabolic problems with time-dependent coefficients, Annali della Scuola Normale Superiore di Pisa, 25 (1997), 329-373.  Google Scholar

[12]

Daniele Antonio Di Pietro and Alexandre Ern, Mathematical Aspects of Discontinuous Galer\-Kin Methods, Springer, 2012.  Google Scholar

[13]

Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology, Vol. II, Springer, 1992. Google Scholar

[14]

Alexandre Ern and Jean-Luc Guermond, Theory and Practice of Finite Elements, Springer, 2004.  Google Scholar

[15]

Alexandre Ern and Jean-Luc Guermond, Discontinuous Galerkin methods for Friedrichs' systems. I. General theory, SIAM J. Numer. Anal., 44 (2006), 753-778.  Google Scholar

[16]

Alexandre Ern and Jean-Luc Guermond, Discontinuous Galerkin methods for Friedrichs' systems. II. Second-order elliptic PDEs, SIAM J. Numer. Anal., 44 (2006), 2363-2388.  Google Scholar

[17]

Alexandre Ern and Jean-Luc Guermond, Discontinuous Galerkin methods for Friedrichs' systems. III. Multifield theories with partial coercivity, SIAM J. Numer. Anal., 46 (2008), 776-804.  Google Scholar

[18]

Alexandre Ern, Jean-Luc Guermond and Gilbert Caplain, An intrinsic criterion for the bijectivity of Hilbert operators related to Friedrichs' systems, Comm. Partial Diff. Eq., 32 (2007), 317-341.  Google Scholar

[19]

Kurt O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math., 11 (1958), 333-418.  Google Scholar

[20]

Max Jensen, Discontinuous Galerkin Methods for Friedrichs Systems with Irregular Solutions, Ph.D thesis, University of Oxford, 2004. Google Scholar

[21]

Jacques-Lous Lions and Enrico Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. 1, Springer, 1972. Google Scholar

[22]

François Murat and Luc Tartar, H-convergence, in Séminaire d'Analyse Fonctionnelle et Numérique de l'Université d'Alger, mimeographed notes, 1978. English translation in Topics in the mathematical modelling of composite materials (eds. A. Cherkaev, R. Kohn) Birkhäuser, 1997, 21-43. Google Scholar

[23]

Jeffrey Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187.  Google Scholar

[24]

Sergio Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore, Ann. Scuola Norm. Sup. Pisa, 21 (1967), 657-699.  Google Scholar

[25]

Sergio Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 571-597.  Google Scholar

[26]

Sergio Spagnolo, Convergence of parabolic equations, Bolletino della Unione Matematica Italiana, 14-B (1977), 547-568.  Google Scholar

[27]

Luc Tartar, Problèmes de contrôle des coefficients dans des équations aux dérivées partielles, in Control Theory, Numerical Methods and Computer Systems Modelling, Lecture notes in Economy and Mathematical Systems, Springer Verlag, 107, 420-426, 1975. Google Scholar

[28]

Luc Tartar, Nonlocal effects induced by homogenisation, in Partial Differential Equations and the Calculus of Variation, Essays in Honor of Ennio de Giorgi, II, Birkh\"auser, Boston, 925-938, 1989. Google Scholar

[29]

Luc Tartar, The General Theory of Homogenization: A Personalized Introduction, Springer, 2009. Google Scholar

[30]

Marcus Waurick, On the theory of homogenization of evolutionary equations in Hilbert spaces,, preprint, ().   Google Scholar

[31]

Vasili V. Žikov, Sergei M. Kozlov, Ol'ga A. Oleinik and Ha Tien Ngoan, Homogenization and G-convergence of differential operators, Uspehi Mat. Nauk, 34 (1979), 65-133. Google Scholar

[32]

Vasili V. Žikov, Sergei M. Kozlov and Ol'ga A. Oleinik, On G-convergence of parabolic operators, Russ. Math. Surv., 36 :1 (1981) 9-60. Google Scholar

show all references

References:
[1]

Grégoire Allaire, Shape Optimization by the Homogenization Method, Springer-Verlag, 2002.  Google Scholar

[2]

Nenad Antonić and Krešimir Burazin, Graph spaces of first-order linear partial differential operators, Math. Communications, 14 (2009), 135-155.  Google Scholar

[3]

Nenad Antonić and Krešimir Burazin, Intrinsic boundary conditions for Friedrichs systems, Comm. Partial Diff. Eq., 35 (2010), 1690-1715.  Google Scholar

[4]

Nenad Antonić and Krešimir Burazin, Boundary operator from matrix field formulation of boundary conditions for Friedrichs systems, J. Diff. Eq., 250 (2011), 3630-3651,  Google Scholar

[5]

Nenad Antonić, Krešimir Burazin and Marko Vrdoljak, Second-order equations as Friedrichs systems, Nonlinear Analysis: RWA, 14 (2014), 290-305.  Google Scholar

[6]

Nenad Antonić, Krešimir Burazin and Marko Vrdoljak, Heat equation as a Friedrichs system, Journal of Mathematical Analysis and Applications, 404 (2013), 537-553.  Google Scholar

[7]

Nenad Antonić and Marko Vrdoljak, Parabolic H-convergence and small-amplitude homogenization, Applicable Analysis, 88 (2009), 1493-1508.  Google Scholar

[8]

Alain Bensoussan, Jacques-Lous Lions and Georgios Papanicolaou, Asymptotic Analysis in Periodic Structures, Northćolland, Amsterdam, 1978. Google Scholar

[9]

Krešimir Burazin, Contributions to the theory of Friedrichs' and Hyperbolic Systems, Ph.D thesis (in Croatian), University of Zagreb, 2008, http://www.mathos.hr/~kburazin/papers/teza.pdf. Google Scholar

[10]

Tan Bui-Thanh, Leszek Demkowicz and Omar Ghattas, A unified discontinuous Petrov-Galerkin method and its analysis for Friedrichs' systems, SIAM J. Numer. Anal., 51 (2013), 1933-1958.  Google Scholar

[11]

Andrea Dall'Aglio and Francois Murat, A corrector result for $H$-converging parabolic problems with time-dependent coefficients, Annali della Scuola Normale Superiore di Pisa, 25 (1997), 329-373.  Google Scholar

[12]

Daniele Antonio Di Pietro and Alexandre Ern, Mathematical Aspects of Discontinuous Galer\-Kin Methods, Springer, 2012.  Google Scholar

[13]

Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology, Vol. II, Springer, 1992. Google Scholar

[14]

Alexandre Ern and Jean-Luc Guermond, Theory and Practice of Finite Elements, Springer, 2004.  Google Scholar

[15]

Alexandre Ern and Jean-Luc Guermond, Discontinuous Galerkin methods for Friedrichs' systems. I. General theory, SIAM J. Numer. Anal., 44 (2006), 753-778.  Google Scholar

[16]

Alexandre Ern and Jean-Luc Guermond, Discontinuous Galerkin methods for Friedrichs' systems. II. Second-order elliptic PDEs, SIAM J. Numer. Anal., 44 (2006), 2363-2388.  Google Scholar

[17]

Alexandre Ern and Jean-Luc Guermond, Discontinuous Galerkin methods for Friedrichs' systems. III. Multifield theories with partial coercivity, SIAM J. Numer. Anal., 46 (2008), 776-804.  Google Scholar

[18]

Alexandre Ern, Jean-Luc Guermond and Gilbert Caplain, An intrinsic criterion for the bijectivity of Hilbert operators related to Friedrichs' systems, Comm. Partial Diff. Eq., 32 (2007), 317-341.  Google Scholar

[19]

Kurt O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math., 11 (1958), 333-418.  Google Scholar

[20]

Max Jensen, Discontinuous Galerkin Methods for Friedrichs Systems with Irregular Solutions, Ph.D thesis, University of Oxford, 2004. Google Scholar

[21]

Jacques-Lous Lions and Enrico Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. 1, Springer, 1972. Google Scholar

[22]

François Murat and Luc Tartar, H-convergence, in Séminaire d'Analyse Fonctionnelle et Numérique de l'Université d'Alger, mimeographed notes, 1978. English translation in Topics in the mathematical modelling of composite materials (eds. A. Cherkaev, R. Kohn) Birkhäuser, 1997, 21-43. Google Scholar

[23]

Jeffrey Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187.  Google Scholar

[24]

Sergio Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore, Ann. Scuola Norm. Sup. Pisa, 21 (1967), 657-699.  Google Scholar

[25]

Sergio Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Ann. Scuola Norm. Sup. Pisa, 22 (1968), 571-597.  Google Scholar

[26]

Sergio Spagnolo, Convergence of parabolic equations, Bolletino della Unione Matematica Italiana, 14-B (1977), 547-568.  Google Scholar

[27]

Luc Tartar, Problèmes de contrôle des coefficients dans des équations aux dérivées partielles, in Control Theory, Numerical Methods and Computer Systems Modelling, Lecture notes in Economy and Mathematical Systems, Springer Verlag, 107, 420-426, 1975. Google Scholar

[28]

Luc Tartar, Nonlocal effects induced by homogenisation, in Partial Differential Equations and the Calculus of Variation, Essays in Honor of Ennio de Giorgi, II, Birkh\"auser, Boston, 925-938, 1989. Google Scholar

[29]

Luc Tartar, The General Theory of Homogenization: A Personalized Introduction, Springer, 2009. Google Scholar

[30]

Marcus Waurick, On the theory of homogenization of evolutionary equations in Hilbert spaces,, preprint, ().   Google Scholar

[31]

Vasili V. Žikov, Sergei M. Kozlov, Ol'ga A. Oleinik and Ha Tien Ngoan, Homogenization and G-convergence of differential operators, Uspehi Mat. Nauk, 34 (1979), 65-133. Google Scholar

[32]

Vasili V. Žikov, Sergei M. Kozlov and Ol'ga A. Oleinik, On G-convergence of parabolic operators, Russ. Math. Surv., 36 :1 (1981) 9-60. Google Scholar

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