Computing and Visualization in Science

, Volume 20, Issue 1–2, pp 29–46 | Cite as

Exploiting multilevel Toeplitz structures in high dimensional nonlocal diffusion

  • Christian VollmannEmail author
  • Volker Schulz
Original Article


We present a finite element implementation for the steady-state nonlocal Dirichlet problem with homogeneous volume constraints. Here, the nonlocal diffusion operator is defined as integral operator characterized by a certain kernel function. We assume that the domain is an arbitrary d-dimensional hyperrectangle and the kernel is translation and reflection invariant. Under these assumptions, we carefully analyze the structure of the stiffness matrix resulting from a continuous Galerkin method with \(Q_1\) elements and exploit this structure in order to cope with the curse of dimensionality associated to nonlocal problems. For the purpose of illustration we choose a particular kernel, which is related to space-fractional diffusion and present numerical results in 1d, 2d and for the first time also in 3d.


Nonlocal diffusion Finite element method Translation invariant kernel Multilevel Toeplitz Fractional diffusion 



The first author has been supported by the German Research Foundation (DFG) within the Research Training Group 2126: “Algorithmic Optimization”.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Universitaet TrierTrierGermany

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