A numerical study of the homogeneous elliptic equation with fractional order boundary conditions

TL;DR

This paper investigates numerical methods for solving a homogeneous elliptic equation with fractional order boundary conditions, demonstrating their accuracy and efficiency through two computational approaches and numerical experiments.

Contribution

It introduces and compares two novel numerical algorithms for fractional boundary conditions in elliptic equations, including integral approximation and auxiliary Cauchy problem methods.

Findings

Both methods are accurate and stable in numerical experiments.

The algorithms are efficient for two-dimensional model problems.

Numerical results confirm the effectiveness of the proposed approaches.

Abstract

We consider the homogeneous equation ${\mathcal A} u=0$, where ${\mathcal A}$ is a symmetric and coercive elliptic operator in $H^1(\Omega)$ with $\Omega$ bounded domain in ${{\mathbb R}}^d$. The boundary conditions involve fractional power $\alpha$, $ 0 < \alpha <1$, of the Steklov spectral operator arising in Dirichlet to Neumann map. For such problems we discuss two different numerical methods: (1) a computational algorithm based on an approximation of the integral representation of the fractional power of the operator and (2) numerical technique involving an auxiliary Cauchy problem for an ultra-parabolic equation and its subsequent approximation by a time stepping technique. For both methods we present numerical experiment for a model two-dimensional problem that demonstrate the accuracy, efficiency, and stability of the algorithms.