DEPARTMENT OF MATHEMATICAL SCIENCES

APPLIED AND COMPUTATIONAL MATHEMATICS SEMINAR

**Speaker:**Christian Glusa, Sandia National Labs

**Title: ***
Adaptive FEM for the fractional Laplacian: a priori and a posteriori
error estimates, efficient implementation and multigrid solver
*

**Abstract:**
We explore the connection between fractional order partial differential
equations in two or more spatial dimensions with boundary integral
operators to develop techniques that enable one to efficiently tackle
the integral fractional Laplacian. We develop all of the components
needed to construct an adaptive finite element code that can be used to
approximate fractional partial differential equations, on non-trivial
domains in \(d\geq 1\) dimensions. Our main approach consists of taking
tools that have been shown to be effective for adaptive boundary element
methods and, where necessary, modifying them so that they can be applied
to the fractional PDE case. Improved a priori error estimates are
derived for the case of quasi-uniform meshes which are seen to deliver
sub-optimal rates of convergence owing to the presence of singularities.
Attention is then turned to the development of an a posteriori error
estimate and error indicators which are suitable for driving an adaptive
refinement procedure. We assume that the resulting refined meshes are
locally quasi-uniform and develop efficient methods for the assembly of
the resulting linear algebraic systems and their solution using
iterative methods, including the multigrid method. The storage of the
dense matrices along with efficient techniques for computing the dense
matrix vector products needed for the iterative solution is also
considered. Importantly, the approximation does not make any strong
assumptions on the shape of the underlying domain and does not rely on
any special structure of the matrix that could be exploited by fast
transforms. The performance and efficiency of the resulting algorithm is
illustrated for a variety of examples.
This is joint work with Mark Ainsworth, Brown University.

**Time:** Friday, October 27, 2017, 1:30-2:30pm

**Place:** Exploratory Hall, Room 4106

Department of Mathematical Sciences

George Mason University

4400 University Drive, MS 3F2

Fairfax, VA 22030-4444

http://math.gmu.edu/

Tel. 703-993-1460, Fax. 703-993-1491