The focus of my research has changed over my career. By training (read: graduate program) is in functional analysis; namely, operator theory on bounded linear operators. For my dissertaion and subsequent research I studied generalized inverses and Drazin inverses in various types of Banach algebras. In these algebras I characterized when operators would have a generlized or Drazin inverse.

Starting in 2005, I began to collaborate with with Dr. Christos Xenophontos who is now at the University of Cyprus by applying functional analysis to numerical analysis. There are certain systems of differential equations that are used to model physical phenomena in engineering that the theory tells us solutions exist, but due to the complexity of the equations exact solutions are extremely difficult if not impossible to find. Thus numerical methods need to be used to find approximate solutions. My current research is to prove that using finite element methods are not only appropriate for these equations but better (more accurate) than other methods.

## Recent Publications

Note: Copyrights to the following papers are held by the publishers. The PDF versions provided here are reprints or preprints. Please treat this material in a way consistent with the "fair use" provisions of appropriate copyright laws.

- Melenk J.M., Xenophontos C., & Oberbroeckling L. (2013).

Analytic regularity for a singular perturbed system of reaction-diffusion equations with multiple scales.

Advances in Computational Mathematics, Vol. 39, 367-394. doi: 10.1007/s10444-012-9284-x. (PDF) - Xenophontos C., Melenk J.M., Madden N., Oberbroeckling L., Panaseti P., & Zouvani Z. (2013).

Hp finite element methods for fourth order singularly perturbed boundary value problems.

Lecture Notes in Computer Science, Vol. 8236, 532 - 539. (PDF) - Melenk J.M., Xenophontos C., & Oberbroeckling L. (2013).

Robust exponential convergence of hp FEM for singularly perturbed reaction-diffusion systems with multiple scales.

IMA Journal of Numerical Analysis, 33, 609-628. doi: 10.1093/imanum/drs013. (Abstract) (Paper) - Melenk J.M., Xenophontos C., & Oberbroeckling L. (2011).

Analytic regularity for a singular perturbed system of reaction-diffusion equations with multiple scales: proofs

posted to arXiv.org (PDF) - Xenophontos C. & Oberbroeckling L. (2010).

On the hp finite element approximation of systems of reaction-diffusion equations by p/hp methods.

Journal of Computational Mathematics Vol. 28, No. 3, 386-400. (PDF) - Oberbroeckling, L.A. (2008).

Drazin inverses in Jörgens algebras of bounded linear operators.

Mathematical Proceedings of the Royal Irish Academy. 108A, 81-87. (PDF) - Xenophontos C. & Oberbroeckling L. (2007).

An hp finite element method for singularly perturbed systems of reaction-diffusion equations.

PAMM - Proc. Appl. Math. Mech. 7, 2020055-2020056 (PDF) - Xenophontos C. & Oberbroeckling L. (2007).

A numerical study on the finite element solution of singularly perturbed systems of reaction-diffusion problems.

Applied Mathematics and Computation Vol 187, 1351 - 1367. (PDF) - Oberbroeckling L. (2006).

Generalised inverses in Jörgens algebras of bounded linear operators.

Mathematical Proceedings of the Royal Irish Academy 106A (1), 85-95. (PDF)