Multigrid methods and Adaptive Finite Element Methods for fractional Laplacian equations

  • Speaker(s)Long Chen (University of California at Irvine)
  • DateFrom 2014-06-26 To 2014-06-26
  • VenueRoom 77201 at #78 courtyard, Beijing International Center for Mathematical Research

Speaker: Long Chen (University of California at Irvine)

Time: Thu, 06/26/2014 - 14:00

Place: Room 77201 at #78 courtyard, Beijing International Center for Mathematical Research

Abstract: In this talk, we will present fast multilevel and adaptive finite element methods for the approximate solution of the discrete problems that arise from the discretization of fractional Laplacian. The fractional Laplacian is a nonlocal operator. To localize it, we solve a Dirichlet to Neumann-type operator via an extension problem. However, this comes at the expense of incorporating one more dimension to the problem, thus motivates our study of adaptive multilevel methods.

We use the multilevel framework developed by Xu and Zikatanov to show nearly uniform convergence of a multilevel method for a class of general degenerate elliptic equations. Because of the singularity of the solution, anisotropic elements in the extended variable are needed in order to obtain quasi-optimal error estimates. For this reason, we also consider a multigrid method with a line smoother and obtain nearly uniform convergence rates.

We derive a computable a posteriori error estimator for the extension problem. Our a posteriori error estimator relies on the solution of small discrete problems on anisotropic cylindrical stars. Under certain assumptions, it exhibits built-in flux equilibration and is equivalent to the energy error up to data oscillation. We design a simple adaptive algorithm and present numerical experiments which reveal a competitive performance.