Mathematisches Kolloquium

 

Freitag, 17.04.2015 14:15 Uhr

Title: Asymptotically compatible discretizations of fractional Laplacian and other nonlocal diffusion problems on bounded domains

Max Gunzburger (Florida State University)

Abstract:
We are interested in the approximation of a class of nonlocal diffusion equations and fractional Laplacian equations on bounded domains. The former are characterized by a horizon parameter measuring the range of nonlocal interactions whereas the latter involve global interactions. We show that conforming Galerkin finite element approximations of the nonlocal diffusion equation are always asymptotically compatible schemes for the corresponding fractional Laplacian model as the value of horizon increases. If the fractional order is less that 1/2, any discontinuous or continuous finite element spaces lead to asymptotically compatible schemes.
It is greater or equal toe 1/2 and less than 1, only continuous finite element spaces are conforming for the corresponding nonlocal equations and result in asymptotically compatible schemes.
The results are developed with minimal regularity assumptions on the solution and are applicable to general geometric meshes with no restrictions on the space dimension. They lead to the convergence of conforming numerical approximations of the nonlocal diffusion models  to the limiting fractional diffusion model when both the mesh size is reduced and the nonlocal interaction range is enlarged, regardless of how these parameters may or may not be dependent.
Furthermore, our results prove the convergence of numerical solutions on a fixed mesh with increasing nonlocal interaction.

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Ort:  Raum 008/SeMath, Pontdriesch 14-16, 52062 Aachen