# Efficient solutions for nonlocal diffusion problems via boundary-adapted spectral methods

## TL;DR

This paper presents a boundary-adapted spectral method for nonlocal diffusion problems that efficiently handles arbitrary boundary conditions, transforming convolution into multiplication in Fourier space, and achieves high accuracy with scalable computations.

## Contribution

The authors develop a spectral method that incorporates volume penalization to handle arbitrary boundary conditions in nonlocal diffusion problems, overcoming limitations of traditional spectral methods.

## Key findings

- Method achieves O(NlogN) computational complexity.
- High accuracy with Dirichlet and Neumann boundary conditions.
- Convergence studies confirm effectiveness and scalability.

## Abstract

We introduce an efficient boundary-adapted spectral method for peridynamic diffusion problems with arbitrary boundary conditions. The spectral approach transforms the convolution integral in the peridynamic formulation into a multiplication in the Fourier space, resulting in computations that scale as O(NlogN). The limitation of regular spectral methods to periodic problems is eliminated using the volume penalization method. We show that arbitrary boundary conditions or volume constraints can be enforced in this way to achieve high levels of accuracy. To test the performance of our approach we compare the computational results with analytical solutions of the nonlocal problem. The performance is tested with convergence studies in terms of nodal discretization and the size of the penalization parameter in problems with Dirichlet and Neumann boundary conditions.

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Source: https://tomesphere.com/paper/1905.03875