G-convergence of Friedrichs systems revisited

We revisit homogenisation theory for Friedrichs systems. In particular, we show that $G$-compactness can be obtained under severely weaker assumptions than in the original work of Burazin and Vrdoljak (2014). In this way we extend the applicability of $G$-compactness results for Friedrichs systems to equations that yield memory effects in the homogenised limit and detour any usage of compactness techniques previously employed.

[1]  S. Trostorff,et al.  A Note on Some Non-Local Boundary Conditions and their Use in Connection with Beltrami Fields , 2023, 2303.10984.

[2]  M. Waurick.  Nonlocal $H$-convergence for topologically nontrivial domains , 2022, 2204.12315.

[3]  Marko Erceg,et al.  Classification of classical Friedrichs differential operators: One-dimensional scalar case , 2022, Communications on Pure and Applied Analysis.

[4]  Christian Seifert,et al.  Evolutionary Equations. , 2020, 2003.12403.

[5]  M. Waurick,et al.  Two-scale homogenization of abstract linear time-dependent PDEs , 2019, Asymptotic Analysis.

[6]  M. Waurick,et al.  Dissipative extensions and port-Hamiltonian operators on networks , 2019, Journal of Differential Equations.

[7]  M. Waurick.  Nonlocal H-convergence , 2018, Calculus of Variations and Partial Differential Equations.

[8]  Marko Erceg,et al.  Friedrichs systems in a Hilbert space framework: Solvability and multiplicity , 2017 .

[9]  M. Waurick,et al.  Some remarks on the notions of boundary systems and boundary triple(t)s , 2017, Mathematische Nachrichten.

[10]  Marko Erceg,et al.  Complex Friedrichs systems and applications , 2017 .

[11]  A. T. Elst,et al.  The Dirichlet-to-Neumann operator for divergence form problems , 2017, Annali di Matematica Pura ed Applicata (1923 -).

[12]  M. Waurick.  G‐convergence and the weak operator topology , 2016 .

[13]  M. Waurick.  On the continuous dependence on the coefficients of evolutionary equations , 2016, 1606.07731.

[14]  Marko Erceg,et al.  Non-Stationary Abstract Friedrichs Systems , 2016 .

[15]  B. Després,et al.  Dissipative formulation of initial boundary value problems for Friedrichs’ systems , 2016 .

[16]  Tan Bui-Thanh,et al.  From Godunov to a unified hybridized discontinuous Galerkin framework for partial differential equations , 2015, J. Comput. Phys..

[17]  K. Burazin,et al.  Homogenisation theory for Friedrichs systems , 2013 .

[18]  M. Waurick.  G-convergence of linear differential equations , 2013, 1302.7207.

[19]  M. Waurick.  A Hilbert space approach to homogenization of linear ordinary differential equations including delay and memory terms , 2012 .

[20]  M. Waurick.  On the homogenization of partial integro-differential-algebraic equations , 2012, 1204.3768.

[21]  Miguel A. Fernández,et al.  Explicit Runge-Kutta Schemes and Finite Elements with Symmetric Stabilization for First-Order Linear PDE Systems , 2010, SIAM J. Numer. Anal..

[22]  Krešimir Burazin,et al.  Intrinsic Boundary Conditions for Friedrichs Systems , 2010 .

[23]  Luc Tartar,et al.  The General Theory of Homogenization: A Personalized Introduction , 2009 .

[24]  R. Picard.  A structural observation for linear material laws in classical mathematical physics , 2009 .

[25]  Nenad Antonić,et al.  Graph spaces of first-order linear partial differential operators , 2009 .

[26]  Jean-Luc Guermond,et al.  An Intrinsic Criterion for the Bijectivity of Hilbert Operators Related to Friedrich' Systems , 2007 .

[27]  Jean-Luc Guermond,et al.  Discontinuous Galerkin Methods for Friedrichs' Systems. I. General theory , 2006, SIAM J. Numer. Anal..

[28]  P. Donato,et al.  An introduction to homogenization , 2000 .

[29]  Gilles A. Francfort,et al.  Homogenization and optimal bounds in linear elasticity , 1986 .

[30]  Jacques-Louis Lions,et al.  Remarks on homogenization , 1985 .

[31]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[32]  Kurt Friedrichs,et al.  Symmetric positive linear differential equations , 1958 .

[33]  Jürgen Fuhrmann,et al.  Guermond : " Theory and Practice of Finite Elements " , 2017 .

[34]  Marcus Waurick,et al.  Homogenization of a class of linear partial differential equations , 2013, Asymptot. Anal..

[35]  M. Waurick.  Limiting Processes in Evolutionary Equations - A Hilbert Space Approach to Homogenization , 2011 .

[36]  Max Jensen,et al.  Discontinuous Galerkin methods for Friedrichs systems with irregular solutions , 2005 .

[37]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[38]  S. Spagnolo,et al.  CONVERGENCE IN ENERGY FOR ELLIPTIC OPERATORS , 1976 .

[39]  Luc Tartar,et al.  Problemes de Controle des Coefficients Dans des Equations aux Derivees Partielles , 1975 .

[40]  S. Spagnolo,et al.  Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore , 1967 .