Communications in Mathematical Sciences

Volume 16 (2018)

Number 5

Fractional operators with inhomogeneous boundary conditions: analysis, control, and discretization

Pages: 1395 – 1426

DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n5.a11

Authors

Harbir Antil (Department of Mathematical Sciences, George Mason University, Fairfax, Virginia, U.S.A.)

Johannes Pfefferer (Department of Mathematics, Technical University of Munich, Germany)

Sergejs Rogovs (Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München, Germany)

Abstract

In this paper, we introduce new characterizations of the 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 equations as constraints.

Keywords

new spectral fractional Laplace operator, nonzero boundary conditions, very weak solution, finite element discretization, error estimates

2010 Mathematics Subject Classification

26A33, 35S15, 49K20, 65N12, 65N30, 65R20

Full Text (PDF format)

The work of the first and second author is partially supported by NSF grants DMS-1521590 and DMS-1818772 and Air Force Office of Scientific Research under Award NO: FA9550-19-1-0036.

Received 6 September 2017

Received revised 8 June 2018

Accepted 8 June 2018

Published 19 December 2018