A generalised matrix transfer technique for the numerical solution of fractional-in-space partial differential equations

n this paper we present a theoretical basis for the Matrix Transfer Technique for approximating solutions to fractional-in-space partial differential equations. Furthermore, we extend the method to the solution of equations involving complete Bernstein functions of infinitesimal generators of bounded $C_0$ semigroups. We prove that, under appropriate conditions on the right hand side function, the matrix transfer technique converges with the same order as the underlying spatial discretisation. When we extend the matrix transfer technique to finite volume and finite element methods, we find that the resulting discretisations are no longer symmetric with respect to the standard Euclidean inner product, but are instead self-adjoint with respect to a more general inner product on $mathbb{R}^n$. We propose an $M$-Lanczos approximation to $f(A)b$ based on the standard Lanczos algorithm under a different inner product and derive an error bound for this case. A number of case studies are presented to illustrate the theory.