Speaker
Description
Many natural and industrial phenomena exhibit nonlocal behaviour in temporal or spatial dimensions. The former is responsible for processes for which its whole history influences the present state. The latter, on the other hand, indicates that faraway regions of the domain may have some impact on local points. This is useful in describing media of high heterogeneity.
Partial differential equations that are nonlocal involve one or several integral operators that encode this behaviour. For example, Riemann-Liouville or Caputo derivatives are used in temporal direction, while fractional Laplacian or its relatives describe spatial nonlocality. When it comes to numerical methods the discretization of these requires more care than their classical versions. Moreover, it is usually much more expensive, both on CPU and the memory, to conduct simulations involving nonlocal equations.
In this talk we will present several approaches to discretize nonlocal and nonlinear parabolic equations based on finite differences, Galerkin method in space and L1 scheme in time. I will also show rigorous and optimal bounds for these numerical methods.
