FEM on nonuniform meshes for nonlocal Laplacian: Semi-analytic Implementation in One Dimension

TL;DR

This paper introduces a semi-analytic finite element method for nonlocal Laplacian problems on nonuniform meshes in one dimension, enabling explicit computation of stiffness matrix entries and analyzing limiting behaviors.

Contribution

It develops a semi-analytic FEM approach for nonlocal Laplacian on nonuniform meshes, bridging to classical FEM in limiting cases, and is the first of its kind in one dimension.

Findings

Explicit integral expressions for stiffness matrix entries.

Automatic transition to classical FEM in limiting cases.

Numerical experiments validating the approach.

Abstract

In this paper, we compute stiffness matrix of the nonlocal Laplacian discretized by the piecewise linear finite element on nonuniform meshes, and implement the FEM in the Fourier transformed domain. We derive useful integral expressions of the entries that allow us to explicitly or semi-analytically evaluate the entries for various interaction kernels. Moreover, the limiting cases of the nonlocal stiffness matrix when the interactional radius $\delta\rightarrow0$ or $\delta\rightarrow\infty$ automatically lead to integer and fractional FEM stiffness matrices, respectively, and the FEM discretisation is intrinsically compatible. We conduct ample numerical experiments to study and predict some of its properties and test on different types of nonlocal problems. To the best of our knowledge, such a semi-analytic approach has not been explored in literature even in the one-dimensional case.