Numerical solution of time-dependent problems with fractional power elliptic operator
This work addresses the computational challenge of solving unsteady fractional PDEs for researchers in numerical analysis and applied mathematics, but the results appear incremental.
The paper develops numerical methods for solving time-dependent fractional power elliptic equations using finite element spatial discretization and two-level time schemes, with Pade-type approximations for the fractional operator. Numerical experiments on a 2D test problem demonstrate the approach.
An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves a fractional power of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, standard two-level schemes are used. The approximate solution at a new time-level is obtained as a solution of a discrete problem with the fractional power of the elliptic operator. A Pade-type approximation is constructed on the basis of special quadrature formulas for an integral representation of the fractional power elliptic operator using explicit schemes. A similar approach is applied in the numerical implementation of implicit schemes. The results of numerical experiments are presented for a test two-dimensional problem.