A finite element method for quantum graphs

被引:29
作者
Arioli, Mario [1 ,2 ]
Benzi, Michele [1 ]
机构
[1] Emory Univ, Dept Math & Comp Sci, Atlanta, GA 30322 USA
[2] CERFACS, Toulouse, France
基金
英国工程与自然科学研究理事会; 美国国家科学基金会;
关键词
quantum graphs; finite element method; complex graphs; sparse matrices; iterative methods; diffusion on graphs; exponential integrators; NETWORKS; ALGORITHM; MATRIX; FLOW;
D O I
10.1093/imanum/drx029
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We study the numerical solution of boundary and initial value problems for differential equations posed on graphs or networks. The graphs of interest are quantum graphs, i.e., metric graphs endowed with a differential operator acting on functions defined on the graph's edges with suitable side conditions. We describe and analyse the use of linear finite elements to discretize the spatial derivatives for a class of linear elliptic model problems. The solution of the discrete equations is discussed in detail in the context of a (nonoverlapping) domain decomposition approach. For model elliptic problems and a wide class of graphs, we show that a combination of Schur complement reduction and diagonally preconditioned conjugate gradients results in optimal complexity. For problems of parabolic type, we consider the use of exponential integrators based on Krylov subspace methods. Numerical results are given for both simple and complex graph topologies.
引用
收藏
页码:1119 / 1163
页数:45
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