4,351 research outputs found
Field representation for optical defect resonances in multilayer microcavities using quasi-normal modes
Quasi-normal modes are used to characterize transmission resonances in 1D optical defect cavities and the related field approximations. We specialize to resonances inside the bandgap of the periodic multilayer mirrors that enclose the defect cavities. Using a template with the most relevant QNMs a variational principle permits to represent the field and the spectral transmission close to resonances
Time-parallel iterative solvers for parabolic evolution equations
We present original time-parallel algorithms for the solution of the implicit
Euler discretization of general linear parabolic evolution equations with
time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory
of parabolic problems, we show that the standard nonsymmetric time-global
system can be equivalently reformulated as an original symmetric saddle-point
system that remains inf-sup stable with respect to the same natural parabolic
norms. We then propose and analyse an efficient and readily implementable
parallel-in-time preconditioner to be used with an inexact Uzawa method. The
proposed preconditioner is non-intrusive and easy to implement in practice, and
also features the key theoretical advantages of robust spectral bounds, leading
to convergence rates that are independent of the number of time-steps, final
time, or spatial mesh sizes, and also a theoretical parallel complexity that
grows only logarithmically with respect to the number of time-steps. Numerical
experiments with large-scale parallel computations show the effectiveness of
the method, along with its good weak and strong scaling properties
Reliable a-posteriori error estimators for -adaptive finite element approximations of eigenvalue/eigenvector problems
We present reliable a-posteriori error estimates for -adaptive finite
element approximations of eigenvalue/eigenvector problems. Starting from our
earlier work on adaptive finite element approximations we show a way to
obtain reliable and efficient a-posteriori estimates in the -setting. At
the core of our analysis is the reduction of the problem on the analysis of the
associated boundary value problem. We start from the analysis of Wohlmuth and
Melenk and combine this with our a-posteriori estimation framework to obtain
eigenvalue/eigenvector approximation bounds.Comment: submitte
A computer-assisted existence proof for Emden's equation on an unbounded L-shaped domain
We prove existence, non-degeneracy, and exponential decay at infinity of a
non-trivial solution to Emden's equation on an unbounded
-shaped domain, subject to Dirichlet boundary conditions. Besides the direct
value of this result, we also regard this solution as a building block for
solutions on expanding bounded domains with corners, to be established in
future work. Our proof makes heavy use of computer assistance: Starting from a
numerical approximate solution, we use a fixed-point argument to prove
existence of a near-by exact solution. The eigenvalue bounds established in the
course of this proof also imply non-degeneracy of the solution
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