Appl Math Optim (2011) 64:313–338
Spectral Solutions of Self-adjoint Elliptic Problems
with Immersed Interfaces
G. Auchmuty · P. K lou
Published online: 29 June 2011
© Springer Science+Business Media, LLC 2011
Abstract This paper describes a spectral representation of solutions of self-adjoint
elliptic problems with immersed interfaces. The interface is assumed to be a simple
non-self-intersecting closed curve that obeys some weak regularity conditions. The
problem is decomposed into two problems, one with zero interface data and the other
with zero exterior boundary data. The problem with zero interface data is solved by
standard spectral methods. The problem with non-zero interface data is solved by
introducing an interface space H
() and constructing an orthonormal basis of this
space. This basis is constructed using a special class of orthogonal eigenfunctions
analogously to the methods used for standard trace spaces by Auchmuty (SIAM J.
Math. Anal. 38, 894–915, 2006). Analytical and numerical approximations of these
eigenfunctions are described and some simulations are presented.
Keywords Immersed interface · Interface eigenproblem · Interface trace space
The paper describes the construction of solutions of self-adjoint second order linear
elliptic equations where the unknown function is required to take speciﬁc values on
Communicating Editor: Irena Lasiecka.
The ﬁrst author was supported in part by NSF grant DMS-0808115. The second author was
supported by the grant MEXC-CT-2005-023843 from the Commission of the European Union.
G. Auchmuty (
Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA
P. Klo u
Institut de Mathèmatiques, Universitè de Neuchâtel, Rue Emile Argand 11, 2007 Neuchâtel,