Isogeometric Collocation Method for the Fractional Laplacian in the 2D Bounded Domain

TL;DR

This paper develops an isogeometric collocation method for solving fractional PDEs involving the fractional Laplacian in 2D, demonstrating higher accuracy and convergence rates compared to finite element methods.

Contribution

The paper introduces a novel isogeometric collocation approach for fractional Laplacian problems, showing improved accuracy and convergence in 2D nonlocal PDEs.

Findings

Monotonous convergence at rate O(N^{-1})

Higher accuracy per degree of freedom than finite element methods

Effective for fractional Poisson and time-dependent fractional porous media equations

Abstract

We consider the isogeometric analysis for fractional PDEs involving the fractional Laplacian in two dimensions. An isogeometric collocation method is developed to discretize the fractional Laplacian and applied to the fractional Poisson problem and the time-dependent fractional porous media equation. Numerical studies exhibit monotonous convergence with a rate of $\mathcal{O}(N^{-1})$, where $N$ is the degrees of freedom. A comparison with finite element analysis shows that the method enjoys higher accuracy per degree of freedom and has a better convergence rate. We demonstrate that isogeometric analysis offers a novel and promising computational tool for nonlocal problems.