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) Optimal solver for the Dirichlet biharmonic equation.
The biharmonic equation is much harder to solve than second-order equations because of severe ill-conditioning.
The fundamental work of Glowinski and Pironneau shows that the biharmonic equation L^2 u = f on R^n, n >= 1, with homogeneous Dirichlet
boundary conditions can be written as two Poisson problems:
L u = w, L w =f, where L is minus Laplacian. The boundary conditions are u = 0 and w= g with M g = p for some pseudodifferential operator M
and known boundary function p.
We have come up with an optimal preconditioner for M and when combined with an optimal Poisson solver, the biharmonic solver is optimal -
meaning that its complexity is proportional to the number of unknowns.
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.
Domain Decomposition Methods
1) An optimized Schwarz method for domains with an arbitrary interface, J. Comput. Applied Math., 235 (2010), pp. 301-314.
2) An optimized Schwarz method for PDEs with discontinuous coefficients (with O. Dubois), Numer. Algorithms., 51 (2009), pp. 115-131.
3) A Lions nonoverlapping domain decomposition method for domains with an arbitrary interface, IMA J. Numer. Anal., DOI:10.1093/imanum/drm011 (2008).
4) On Linear Monotone and Schwarz Alternating Methods for Nonlinear Elliptic PDEs, Numer. Math., 93 (2002), pp. 109-129.
5) On Schwarz Alternating Methods for the Incompressible Navier Stokes Equation, SIAM J.S.C., 22 (2001), pp. 1974-1986.
6) On Schwarz Alternating Methods for Nonlinear Elliptic PDEs, SIAM J.S.C., 21 (2000), pp.1506-1523.
1) On an optimal biharmonic solver, preprint.
2) An Optimal Solver for Fourth-Order PDEs (with J. Lippuner), preprint.
1) Pseudospectral Mapping Theorem II, ETNA, 38, (2011), pp.168--183.
2) A Pseudospectral Mapping Theorem, Math. Comp., 72 (2003), pp. 1841-1854.
3) Computation of Pseudospectra by Continuation, SIAM J.S.C., 18 (1997), pp. 565-573.
Spectral methods for PDEs on complex geometry
1) Spectral Domain Imbedding for Elliptic PDEs in Complex Geometry, J. Comput. and Appl. Math. (2008), DOI:10.1016/j.cam.2008.08.034.
Numerical method for PDEs on domains with a cusp
1) A Numerical Study of the Dirichlet and Neumann Eigenvalue Problem of the Laplacian on Cusp Domains, preprint.