On a Framework for the Stability and Convergence Analysis of Discrete Schemes for Nonstationary Nonlocal Problems of Parabolic Type

The main aim of this article is to propose a general framework for the theoretical analysis of discrete schemes used to solve multi-dimensional parabolic problems with fractional power elliptic operators. This analysis is split into three parts. The first part is based on techniques well developed for the solution of nonlocal elliptic problems. The obtained discrete elliptic operators are used to formulate semi-discrete approximations. Next, the fully discrete schemes are constructed by applying the classical and robust approximations of time derivatives. The existing stability and convergence results are directly included in the new framework. In the third part, approximations of transfer operators are constructed by using uniform and the best uniform rational approximations. The stability and accuracy of the obtained local discrete schemes are investigated. The results of computational experiments are presented and analyzed. A three-dimensional test problem is solved. The rational approximations are constructed by using the BRASIL algorithm.