Exploiting multilevel Toeplitz structures in high dimensional nonlocal diffusion

@article{Vollmann2019ExploitingMT,
  title={Exploiting multilevel Toeplitz structures in high dimensional nonlocal diffusion},
  author={Christian Vollmann and Volker Schulz},
  journal={Computing and Visualization in Science},
  year={2019},
  volume={20},
  pages={29-46}
}
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… 

Efficient quadrature rules for finite element discretizations of nonlocal equations

This work designs efficient quadrature rules for finite element (FE) discretizations of nonlocal diffusion problems with compactly supported kernel functions by introducing a parametrized mollifying function that improves the regularity of the integrand, utilizing an adaptive integration technique, and exploiting parallelization.

Regularity and approximation analyses of nonlocal variational equality and inequality problems

We consider linear and obstacle problems driven by a nonlocal integral operator, for which nonlocal interactions are restricted to a ball of finite radius. These type of operators are used to model

Numerical methods for nonlocal and fractional models

This article considers a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis, and specific concrete examples, and extensive discussions about numerical methods for determining approximate solutions of the nonlocal models considered.

A cookbook for finite element methods for nonlocal problems, including quadrature rules and approximate Euclidean balls

This work focuses on developing recipes for the efficient assembly of FEM stiffness matrices and on the choice of quadrature rules for the double integrals that contribute to the assembly efficiency and also posses sufficient accuracy.

References

SHOWING 1-10 OF 31 REFERENCES

Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints

It is shown that fractional Laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that the authors consider.

The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator

Regularity and approximation analyses of nonlocal variational equality and inequality problems

We consider linear and obstacle problems driven by a nonlocal integral operator, for which nonlocal interactions are restricted to a ball of finite radius. These type of operators are used to model

A Fast Finite Difference Method for Two-Dimensional Space-Fractional Diffusion Equations

A fast and yet accurate solution method for the implicit finite difference discretization of space-fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient matrices is developed.

Multigrid method for fractional diffusion equations

Convergence proof for the multigrid method of the nonlocal model

This paper provides the detailed proof of the convergence of the two-grid method for the nonlocal model of peridynamics and some special cases of the full multigrid and the V-cycle multigrids are also discussed.

A direct O(N log2 N) finite difference method for fractional diffusion equations

Non-local dirichlet forms and symmetric jump processes

We consider the non-local symmetric Dirichlet form (E,F) given by with F the closure with respect to E 1 of the set of C 1 functions on R d with compact support, where E 1 (f, f):= E(f, f) + f Rd