## Faculty members involved

**Shaun Lui** (link).
My research in numerical PDEs revolves around domain decomposition methods (DDMs). DDMs are parallel iterative methods for solving PDEs and are optimal (in terms of efficiency) for certain classes of PDEs. The idea is to subdivide the domain into overlapping or non-overlapping subdomains, solve a sequence of PDEs on the subdomains, and then paste together the subdomain solutions to form a global solution.

Another research direction involves space-time spectral methods. Classical spectral methods for time dependent PDEs use low-order discretization of the time derivative and spectral discretization of the spatial derivatives, leading to slow convergence due to temporal errors. Space-time spectral methods apply spectral discretization in both space and time resulting in fully spectral convergence.

My other research area deals with pseudospectrum, which is a generalization of the spectrum of a matrix (or operator). Pseudospectra are important in the stability theory of differential equations, for instance, and have found applications in many areas of mathematical sciences.

**Felicia Magpantay** (link).
The use of delay differential equations (DDEs) has been growing in popularity, especially in control systems engineering and mathematical biology. Simple adaptations of standard numerical methods for ordinary differential equations can fail when applied to DDEs, especially when the delay is state-dependent. In this group we work on developing stable schemes for numerically solving general DDEs, and proving the robustness of these methods using techniques for the analytic solutions.

**Richard Slevinsky** (link).
Singular integral equations arise in the reformulation of elliptic partial differntial equations where data is defined on prescribed boundaries of reduced dimensionality. The reduction of dimensionality and transformation from an elliptic partial differential equation into a singular integral eequation arises naturally from Green's representation theorem. Singular integral equations have a rich history in scattering problems for electromagnetics and seismic imaging, fracture mechanics, fluid dynamics, and beam physics. For applications including random matrix theory, asymptotics of orthogonal polynomials, and integrable systems, singular integral equations arise via reformulation as Riemann--Hilbert problems.

In this group, we develop a new class of fast, stable, and well-conditioned spectral methods for singular integral equations. Combination of direct solvers with hierarchical solvers allows numerical simulations with domains consistsing of thousands of disjoint boundaries with millions of degrees of freedom. Use of a hierarchical solver as a pre-conditioner in a parallel iterative solver will extend this to new problems involving millions of disjoint boundaries. To extend the spectral method to more complicated geometries, we develop fast and stable algorithms for transforming expansion coefficients in Chebyshev bases to more exotic polynomial bases. As the spectral method is extended to three-dimensional elliptic partial differential equations, new fast and stable algorithms for transforming expansion coefficients will be required and new applications will also be explored including cloaking, scattering from fractal antennae, and scattering in parabolically stratified media such as optical fibres.

## Current students and postdoctoral fellows

- Nazila Akhavan Kharazian (M.Sc.) - F. Magpantay
- Clifford Allotey (M.Sc.) - F. Magpantay
- Avleen Kaur (M.Sc.) - C. Cowan & S.H. Lui
- Sarah Nataj (Ph.D.) - S.H. Lui

## Representative recent publications

- S. Lui, A Lions nonoverlapping domain decomposition method for domains with an arbitrary interface, IMA J. Numer. Anal., 29, (2009), pp.332-349
- S. Lui, Numerical Analysis of PDEs, Wiley (2011)
- S. Lui, Pseudospectral Mapping Theorem II, ETNA, 38, (2011), pp.168-183
- S. Lui, Legendre Spectral Collocation in Space and Time for PDEs, Numer. Math. (2016)
- F.M.G. Magpantay, N. Kosovalic and J. Wu (2014) An age-structured population model with state-dependent delay: derivation and numerical integration. SIAM J. Numer. Anal., 52(2): 735-756.
- R. M. Slevinsky and S. Olver (2015) A fast and well-conditioned spectral method for singular integral equations. arXiv:1507.00596.
- R. M. Slevinsky and S. Olver (2015) On the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc collocation method. SIAM J. Sci. Comput., 37:A676--A700.
- P. Gaudreau, R. M. Slevinsky and H. Safouhi (2015) Computing energy eigenvalues of anharmonic oscillators using the double exponential Sinc collocation method. Ann. Phys., 360:520--538.