Exploiting multilevel Toeplitz structures in high dimensional nonlocal diffusion

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 $$Q_1$$Q1 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.

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