Spectral Solutions of Self-adjoint Elliptic Problems with Immersed Interfaces

Spectral Solutions of Self-adjoint Elliptic Problems with Immersed Interfaces 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 Γ (Ω) 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Spectral Solutions of Self-adjoint Elliptic Problems with Immersed Interfaces

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Publisher
Springer-Verlag
Copyright
Copyright © 2011 by Springer Science+Business Media, LLC
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics; Numerical and Computational Physics; Mathematical Methods in Physics; Systems Theory, Control
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-011-9142-0
Publisher site
See Article on Publisher Site

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 Γ (Ω) 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.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Dec 1, 2011

References

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