| Xiaochuan Tian, Department of Mathematics, Columbia University |
| Title: Asymptotically Compatible Schemes for Robust Discretization of Nonlocal Models |
| Abstract:Many problems in nature, being characterized by a parameter, are of interests both with a fixed parameter value and with the parameter approaching an asymptotic limit. Numerical schemes that are convergent in both regimes offer robust discretizations which can be highly desirable in practice. The asymptotically compatible(AC) schemes discussed here meet such objectives for a class of parametrized problems. An abstract mathematical framework is given here together with applications to the numerical solutions of some nonlocal models featured with a horizon parameter which characterizes the nonlocal interaction length. In particular, we will discuss AC schemes for robust discretization of nonlocal diffusion with horizon parameter going to zero or infinity, where the two cases approximates classical diffusion and fractional diffusion respectively. Moreover, a nonconforming DG scheme is proposed for nonlocal models with its convergence established by the theory of AC schemes. |