Publications

We introduce a framework for repurposing error estimators for source problems to compute an estimator for the gap between eigenspaces and their discretizations. Of interest are eigenspaces of finite clusters of eigenvalues of unbounded nonselfadjoint linear operators with compact resolvent. Eigenspaces and eigenvalues of rational functions of such operators are studied as a first step. Under an assumption of convergence of resolvent approximations in the operator norm and an assumption on global reliability of source problem error estimators, we show that the gap in eigenspace approximations can be bounded by a globally reliable and computable error estimator. Also included are applications of the theoretical framework to first-order system least squares (FOSLS) discretizations and discontinuous Petrov-Galerkin (DPG) discretizations, both yielding new estimators for the error gap. Numerical experiments with a selfadjoint model problem and with a leaky nonselfadjoint waveguide eigenproblem show that adaptive algorithms using the new estimators give refinement patterns that target the cluster as a whole instead of individual eigenfunctions.

We introduce an adaptive finite element scheme for the efficient approximation of a (large) collection of eigenpairs of selfadjoint elliptic operators in which the adaptive refinement is driven by the solution of a single source problem --the so-called landscape problem for the operator-- instead of refining based on the computed eigenpairs. Some theoretical justification for the approach is provided, and extensive empirical results indicate that it can provide an attractive alternative to standard adaptive schemes, particularly in the hp-adaptive environment.

An adaptive algorithm for computing eigenmodes and propagation constants of optical fibers is proposed. The algorithm is built using a dual-weighted residual error estimator. The residuals are based on the eigensystem for leaky hybrid modes obtained from Maxwell equations truncated to a finite domain after a transformation by a perfectly matched layer. The adaptive algorithm is then applied to compute practically interesting modes for multiple fiber microstructures. Emerging microstructured optical fibers are characterized by complex geometrical features in their transverse cross-section. Their leaky modes, useful for confining and propagating light in their cores, often exhibit fine scale features. The adaptive algorithm automatically captures these features without any expert input. The results also show that confinement losses of these modes are captured accurately on the adaptively found meshes.

Highly conductive thin casings pose a great challenge in the numerical simulation of well-logging instruments. Witty asymptotic models may replace the presence of casings by impedance transmission conditions in those numerical simulations. The accuracy of such numerical schemes can be tested against benchmark solutions computed semi-analytically in simple geometrical configurations. This paper provides a general approach to construct those benchmark solutions for three different models: one reference model that indeed considers the presence of the casing; one asymptotic model that avoids computations in the casing domain; and one asymptotic model that reduces the presence of the casing to an interface. Our technique uses a Fourier representation of the solutions, where special care has been taken in the analytical integration of singularities to avoid numerical instabilities.