Existence results for general systems of differential equations on one-dimensional networks and prewavelets approximation
Denis Mercier, Serge Nicaise · Discrete and Continuous Dynamical Systems · 1998
In this paper, we first prove existenceresults for general systems of differential equations of parabolic and hyperbolictype in a Hilbert space setting using the notion ofAgmon-Douglis-Nirenberg elliptic systems on a half-line and finding a necessaryand sufficient condition on the boundary and/or transmission conditions which insuresthe dissipativity of the (spatial) operators.Our second goal is to take advantage of the one-dimensional structure ofnetworks in order to build appropriate prewavelet bases in viewto the numerical approximation of the above problems. Indeed weshow that the use of such bases for their approximation (by the Galerkin methodfor elliptic operators and a fully discrete scheme for parabolic ones)leads to linear systems which can be preconditioned by a diagonal matrix andthen can be reduced to systems with a condition number uniformly bounded (withrespect to the mesh parameter).