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

We consider the homogeneous equation $\mathcal{A}u=0$, where $\mathcal{A}$ is a symmetric and coercive elliptic operator in $H^1(\mathrm{\Omega })$ with $\mathrm{\Omega }$ 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.