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April  1998, 4(2): 273-300. doi: 10.3934/dcds.1998.4.273

Existence results for general systems of differential equations on one-dimensional networks and prewavelets approximation

Denis Mercier 1, and  Serge Nicaise 2,

1. 

Université de Valenciennes et du Hainaut Cambrésis, Limav, B.P. 311, 59304 Valenciennes Cedex, France
2. 

Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, F-59313 - Valenciennes Cedex 9, France

Received  April 1997 Revised  June 1997 Published  February 1998

In this paper, we first prove existence results for general systems of differential equations of parabolic and hyperbolic type in a Hilbert space setting using the notion of Agmon-Douglis-Nirenberg elliptic systems on a half-line and finding a necessary and sufficient condition on the boundary and/or transmission conditions which insures the dissipativity of the (spatial) operators. Our second goal is to take advantage of the one-dimensional structure of networks in order to build appropriate prewavelet bases in view to the numerical approximation of the above problems. Indeed we show that the use of such bases for their approximation (by the Galerkin method for elliptic operators and a fully discrete scheme for parabolic ones) leads to linear systems which can be preconditioned by a diagonal matrix and then can be reduced to systems with a condition number uniformly bounded (with respect to the mesh parameter).
Keywords: systems, Networks, differential equations, prewavelets..
Mathematics Subject Classification: 35G10, 35E20, 65N5.
Citation: Denis Mercier, Serge Nicaise. Existence results for general systems of differential equations on one-dimensional networks and prewavelets approximation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 273-300. doi: 10.3934/dcds.1998.4.273
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