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Center for Nonlinear Studies |
We consider the numerical solution of various types of problems for differential equations posed on graphs or networks. More specifically, the talk is concerned with quantum graphs, which are metric graphs endowed with a self-adjoint differential operator (Hamiltonian) acting on functions defined on the graph's edges with suitable side conditions. We describe and analyze the use of a linear finite element method for the spatial discretization of a class of Hamiltonians. The solution of the discrete equations is achieved by means of a (non-overlapping) domain decomposition approach. For model elliptic problems and a wide class of graphs, we show that a combination of Schur complement reduction and the diagonally preconditioned conjugate gradient method results in optimal complexity. We also discuss time-dependent problems of parabolic type and eigenvalue problems if time allows. Numerical results are presented for both simple and complex graph topologies.
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