S. H. Lui

Associate Head (Graduate Studies)
536 Machray Hall
University of Manitoba
Winnipeg, Manitoba
Canada R3T 2N2
(204) 474-9578
B.Sc., M.Sc. (Toronto)
Ph.D. (California Institute of Technology)

Research Areas:

Numerical Analysis, Applied PDEs

1) Domain decomposition methods (DDMs) for PDEs.

DDMs have become an exciting and dynamic research area in scientific computation. The idea is very simple - split up the domain of the problem into many subdomains and then solve the PDE in each subdomain. The global solution is obtained by pasting together the limits of sequences of subdomain solutions. Benefits include parallel computation and isolation of difficult subdomains containing, for instance, boundary layers or geometric singularities. My focus has been on proving convergence of such sequences of subdomain solutions to the global solution of nonlinear PDEs.

Recently, I have been looking into DDMs where Robin or higher order boundary conditions are used along the artificial interface. The main problem is how to optimize the parameters in the boundary conditions for fastest convergence.

2) Pseudospectra.

When a matrix is normal, its eigenvalues determine the stability properties of the system associated with the matrix. When the matrix is not normal, then the eigenvalues are no longer reliable indicators. The pseudospectrum of a matrix is a generalization of the spectrum of a matrix. It gives a quantitative estimate of departure from non-normality and can give some stability information for non-normal matrices. My interests include fast computation of pseudospectra as well as a generalization of the spectral mapping theorem from the spectrum of an operator to its pseudospectrum.

3) Numerical PDEs.

We consider the numerical solution of PDEs on complex domains, possibly with singularities. One recent direction is spectral methods for time dependent PDEs which are spectrally accurate in both space and time for smooth solutions. The classical way is to use a low-order finite difference scheme for the time derivative and spectral scheme for spatial derivatives. This is not ideal since the time discretization error overwhelms the spatial discretization error. One drawback of space-time spectral methods is that time-stepping is no longer possible - unknowns over all time must be solved at the same time. Another line of work involves analysis of numerical methods for fractional PDEs.

4) Nonlinear Elliptic PDEs.

Let L denote the (negative) Laplacian and p>0. Consider positive solutions of L u = u^p on a bounded domain with a homogeneous Dirichlet boundary condition. This nonlinear elliptic PDE and its many variations have played a central in the development of the calculus of variations, bifurcation theory, critical Sobolev exponent, Pohazaev identity, etc. In this research, we consider L u = |D u|^p and study existence, regularity and singularities of solutions. A family of deadcore solutions are found when p<1.

Recent Publications:

Domain Decomposition Methods

7) Adaptive Wavelet Schwarz methods for the Navier-Stokes equation (with S. Dahlke, D. Lellek and R. Stevenson), Numer. Func. Anal. Optim., 37 (2016), pp.1213-1234.
6) An optimized Schwarz method for domains with an arbitrary interface, J. Comput. Applied Math., 235 (2010), pp. 301-314.
5) An optimized Schwarz method for PDEs with discontinuous coefficients (with O. Dubois), Numer. Algorithms., 51 (2009), pp. 115-131.
4) A Lions nonoverlapping domain decomposition method for domains with an arbitrary interface, IMA J. Numer. Anal., DOI:10.1093/imanum/drm011 (2008).
3) On Linear Monotone and Schwarz Alternating Methods for Nonlinear Elliptic PDEs, Numer. Math., 93 (2002), pp. 109-129.
2) On Schwarz Alternating Methods for the Incompressible Navier Stokes Equation, SIAM J.S.C., 22 (2001), pp. 1974-1986.
1) On Schwarz Alternating Methods for Nonlinear Elliptic PDEs, SIAM J.S.C., 21 (2000), pp.1506-1523.


5) Pseudospectrum and condition spectrum (with K. Kumar), Operators & Matrices, 9 (2015), pp. 121-145.
4) On some properties of the pseudo spectral radius (with K. Kumar), Elect. J. Lin. Alg., 27 (2014), pp. 342-353.
3) Pseudospectral Mapping Theorem II, ETNA, 38, (2011), pp.168--183.
2) A Pseudospectral Mapping Theorem, Math. Comp., 72 (2003), pp. 1841-1854.
1) Computation of Pseudospectra by Continuation, SIAM J.S.C., 18 (1997), pp. 565-573.

Numerical PDEs

3) Legendre spectral collocation in space and time for PDEs. Numer. Math., 136 (2017), pp.75-99
2) A Numerical Study of the Dirichlet and Neumann Eigenvalue Problem of the Laplacian on Cusp Domains, J. Comput. Methods in Sci & Eng., 13 (2013) pp. 433-437.
1) Spectral Domain Imbedding for Elliptic PDEs in Complex Geometry, J. Comput. and Appl. Math. 225 (2009), pp541-557.

Nonlinear Elliptic PDEs

2) Singular solutions of elliptic equations involving nonlinear gradient terms on perturbations of the ball (with A. Aghajani and C. Cowan), J. Diff. Eqns, 264 (2018), pp2865-2896.
1) Existence and regularity of solutions of advection problem (with A. Aghajani and C. Cowan), Nonlinear Analysis, 166 (2018), pp. 19-47.

Numerical Analysis of PDEs, Wiley, 2011