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

Abstract

We consider the homogeneous equation ${\mathcal A} u=0$, where ${\mathcal A}$ is a symmetric and coercive elliptic operator in $H^1(Ω)$ with $Ω$ bounded domain in ${\mathbb R}^d$. The boundary conditions involve fractional power $α$, $ 0 < α<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.