2017/03/15 by Harbir Antil, Antil, Harbir, Johannes Pfefferer +3
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1703.05256
openalex publication_date 2017/03/15 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
In this paper we introduce new characterizations of spectral fractional\nLaplacian to incorporate nonhomogeneous Dirichlet and Neumann boundary\nconditions. The classical cases with homogeneous boundary conditions arise as a\nspecial case. We apply our definition to fractional elliptic equations of order\ns \∈ (0,1) with nonzero Dirichlet and Neumann boundary condition. Here the\ndomain \Ω is assumed to be a bounded, quasi-convex Lipschitz domain. To\nimpose the nonzero boundary conditions, we construct fractional harmonic\nextensions of the boundary data. It is shown that solving for the fractional\nharmonic extension is equivalent to solving for the standard harmonic extension\nin the very-weak form. The latter result is of independent interest as well.\nThe remaining fractional elliptic problem (with homogeneous boundary data) can\nbe realized using the existing techniques. We introduce finite element\ndiscretizations and derive discretization error estimates in natural norms,\nwhich are confirmed by numerical experiments. We also apply our\ncharacterizations to Dirichlet and Neumann boundary optimal control problems\nwith fractional elliptic equation as constraints.\n