On Construction of Partially Dimension-Reduced Approximations for Nonstationary Nonlocal Problems of a Parabolic Type

The main aim of this article is to propose an adaptive method to solve multidimensional parabolic problems with fractional power elliptic operators. The adaptivity technique is based on a very efficient method when the multidimensional problem is approximated by a partially dimension-reduced mathematical model. Then in the greater part of the domain, only one-dimensional problems are solved. For the first time such a technique is applied for problems with nonlocal diffusion operators. It is well known that, even for classical local diffusion operators, the averaged flux conjugation conditions become nonlocal. Efficient finite volume type discrete schemes are constructed and analysed. The stability and accuracy of obtained local discrete schemes is investigated. The results of computational experiments are presented and compared with theoretical results.