Stable multi-domain spectral penalty methods for fractional partial differential equations

We introduce stable multi-domain spectral penalty methods suitable for solving fractional partial differential equations with fractional derivatives of any order. The fractional derivative is approximated in each sub domain using orthogonal polynomials and stability of the scheme is achieved through weakly imposed boundary and interface conditions by using a penalty term. We discuss accuracy and stability of the scheme and prove that the accuracy depends on the order of the fractional derivative. The analysis is illustrated through numerical examples, including fractional advection and diffusion problems.