TY - JOUR

T1 - Complex line bundle Laplacians

AU - Vais, Alexander

AU - Berger, Benjamin

AU - Wolter, Franz Erich

PY - 2013/5

Y1 - 2013/5

N2 - In the present work, we extend the theoretical and numerical discussion of the well-known Laplace-Beltrami operator by equipping the underlying manifolds with additional structure provided by vector bundles. Focusing on the particular class of flat complex line bundles, we examine a whole family of Laplacians including the Laplace-Beltrami operator as a special case. To demonstrate that our proposed approach is numerically feasible, we describe a robust and efficient finiteelement discretization, supplementing the theoretical discussion with first numerical spectral decompositions of those Laplacians. Our method is based on the concept of introducing complex phase discontinuities into the finite element basis functions across a set of homology generators of the given manifold. More precisely, given an m-dimensional manifold M and a set of n generators that span the relative homology group Hm-1(M, ∂M), we have the freedom to choose n phase shifts, one for each generator, resulting in a ndimensional family of Laplacians with associated spectra and eigenfunctions. The spectra and absolute magnitudes of the eigenfunctions are not influenced by the exact location of the paths, depending only on their corresponding homology classes. Employing our discretization technique, we provide and discuss several interesting computational examples high- lighting special properties of the resulting spectral decompositions. We examine the spectrum, the eigenfunctions and their zero sets which depend continuously on the choice of phase shifts.

AB - In the present work, we extend the theoretical and numerical discussion of the well-known Laplace-Beltrami operator by equipping the underlying manifolds with additional structure provided by vector bundles. Focusing on the particular class of flat complex line bundles, we examine a whole family of Laplacians including the Laplace-Beltrami operator as a special case. To demonstrate that our proposed approach is numerically feasible, we describe a robust and efficient finiteelement discretization, supplementing the theoretical discussion with first numerical spectral decompositions of those Laplacians. Our method is based on the concept of introducing complex phase discontinuities into the finite element basis functions across a set of homology generators of the given manifold. More precisely, given an m-dimensional manifold M and a set of n generators that span the relative homology group Hm-1(M, ∂M), we have the freedom to choose n phase shifts, one for each generator, resulting in a ndimensional family of Laplacians with associated spectra and eigenfunctions. The spectra and absolute magnitudes of the eigenfunctions are not influenced by the exact location of the paths, depending only on their corresponding homology classes. Employing our discretization technique, we provide and discuss several interesting computational examples high- lighting special properties of the resulting spectral decompositions. We examine the spectrum, the eigenfunctions and their zero sets which depend continuously on the choice of phase shifts.

KW - Complex vector bundles

KW - Finite elements

KW - Laplace operator

KW - Spectral decomposition

UR - http://www.scopus.com/inward/record.url?scp=84893647779&partnerID=8YFLogxK

U2 - 10.1007/s00371-012-0737-5

DO - 10.1007/s00371-012-0737-5

M3 - Article

AN - SCOPUS:84893647779

VL - 29

SP - 345

EP - 357

JO - Visual Computer

JF - Visual Computer

SN - 0178-2789

IS - 5

ER -