Solving Elliptic PDEs on Metric Graphs: Finite Element Discretization, Multigrid Method and PCG Solver

In this monograph, two finite element (FE) methods for solving elliptic partial differential equations (PDEs) on metric graphs are discussed. The first is a multigrid solver, while the second is a PCG method using a domain decomposition approach from [AB]. In order to formulate PDEs on graphs, we work with metric graphs. Metric graphs provide an edgewise parametrization of the graph, such that differential operators can be defined on them. For both methods, we consider an elliptic PDE with Neumann-Kirchhoff conditions. The graph is discretized using a FE discretization with a hat function basis, as described in [AB]. By combining this discretization with a weak formulation of the PDE, we can find an approximation to the solution in the FE discretization space by solving the resulting system of equations. For the multigrid method, I develop suitable intergrid operators for the solution of this system of equations. By adapting the classical multigrid method from [B] to the structure of metric graphs, I prove the convergence of the multigrid method on metric graphs. For the PCG method, the system is decomposed into a Schur-complement system. [AB] used a preconditioner based on a comparability between the Schur-complement and the Laplacian matrix of the underlying graph. This monograph quantifies the comparability, providing bounds dependent solely on the properties of the graph. [AB] M. Arioli, M. Benzi, A finite element method for quantum graphs, IMA Journal of Numerical Analysis, vol. 38, no. 3, pp. 1119-1163, 2017. [B] D. Braess, Finite Elements: Theory, fast solvers, and Applications in Solid Mechanics, 3rd Edition, Cambridge University Press, 2007.