DEPARTMENT OF MATHEMATICAL SCIENCES

APPLIED AND COMPUTATIONAL MATHEMATICS SEMINAR

**Speaker:**Harbir Antil, George Mason University

**Title: ***
Fractional Operators with Inhomogeneous Boundary Conditions: Analysis, Control, and Discretization
*

**Abstract:**
In this talk we introduce new characterizations of spectral fractional
Laplacian to incorporate nonhomogeneous Dirichlet and Neumann boundary
conditions. The classical cases with homogeneous boundary conditions
arise as a special case. We apply our definition to fractional
elliptic equations of order s in (0,1) with nonzero Dirichlet and
Neumann boundary conditions. Here the domain, Omega, is assumed to be a bounded, quasi-convex Lipschitz domain.
To impose the nonzero boundary conditions, we construct fractional harmonic extensions of the boundary data. It is shown that solving for the fractional harmonic extension is equivalent to solving for the standard harmonic extension in the very-weak form. The latter result is of independent interest as well. The remaining fractional elliptic problem (with homogeneous boundary data) can be realized using the existing techniques. We introduce finite element discretizations and derive discretization error estimates in natural norms, which are confirmed by numerical experiments. We also apply our characterizations to Dirichlet and Neumann boundary optimal control problems with fractional elliptic equation as constraints.

**Time:** Friday, April 28, 2017, 1:30-2:20 p.m.

**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