Research
My research interests center around the development and analysis of
algorithms
for the solution of partial differential equations that arise in
engineering
and science. I am particularly interested in bringing new tools to bear
upon
challenging problems that arise in the numerical approximation of
pde's.
I am also interested in the design and analysis of mesh generation
algorithms.
Two dimensional mesh generation has evolved to a very satisfactory
state.
This mesh of New
Zealand was generated by Steven Pav, a former Ph.D. student, who
analyzed the
performance of Ruppert's algorithm when small angles are present in
the input. Subsequently we developed algorithms that could accommodate
curved input constraints containing zero tangent angles, such as
the cusps illustrated in this
mesh.
This past decade has witnessed substantial breakthroughs in the analysis of
three dimensional mesh generation algorithms. Building upon these developments
a former student, Alex Rand, has developed a three dimensional
mesh generation code (dir3) capable of meshing an arbitrary "piecewise
linear complex". A beta release of this code is available at
dir3 website ; you are welcome to download
this code and we welcome comments and/or suggestions on your
experiences using it.
Selected Publications
- (with M. Ondrejat and A. Prohl)
Numerical Approximaion of Nonlinear SPDE's
SPDE: Analysis and Computations, September 2022,
DOI 10.1007/s40072-022-00271-9.
(pdf)
- (with B. Sequin)
Multi-component Multiphase Flow Through a
Poroelastic Medium,
Journal of Elasticity, February 2019.
(pdf)
- (with T. J. Murphy)
Control Volume Approximation of Degenerate Two-Phase Porous Flows,
SINUM, Vol. 57, No. 2, pp. 527-546, March 2019.
(pdf)
(tar.gz)
Corrigendum for the published version of the paper.
(pdf)
- (with B. Sequin)
Multi-component Multiphase Flow,
Archive for Rational Mechancics and Analysis, Vol 23, pp 2171--2196,
March 2020.
(pdf)
- (with L. Perrotti and D. Wang)
Numerical Approximation of Viscoelastic Fluids,
M2AN Math. Model. Numer. Anal. 51 (2017), pp 1119-1144.
(pdf)
- (with C. Calderer, D. Golovaty and O. Lavrentovich)
Modeling of Nematic Electrolytes and Nonlinear
Electroosmosis,
SIAM Journal of Applied Mathematics,
Volume 76, No. 6, (2016), pp 2260-2285.
(pdf)
- (with J. Howell and M. Neilan)
A Dual--Mixed Finite Element Method for the Brinkman Problem,
SMAI Journal of Computational Mathematics,
Volume 2, (2016), pp 1-17.
(pdf)
- High Order Schemes for Wave Equations,
SIAM J. Numer. Anal. 52 (2014), no. 3, pp 1398-1417.
(pdf)
- A $C^1$ Tetrahedral Finite Element Without Edge
Degrees of Freedom,
SIAM J. Numer. Anal. 52 (2014), no. 1, pp 330-342.
(pdf)
The following corrects the statment of Lemma 2.2 of the published
version of the paper.
(pdf)
- (with J. Howell) Dual Mixed Finite Element Methods for the
Navier Stokes Equations,
M2AN 47 (2013) pp 789--805.
(pdf)
- Numerical Approximation of Nematic Liquid Crystal Flows Governed
by the Ericksen Leslie Equations,
M2AN Math. Model. Numer. Anal. 45 (2011), pp 523--540.
(pdf)
- (with J. Howell) Inf-Sup Conditions for Twofold Saddle
Point Problems,
Numer. Math. 118 (2011), no. 4, pp 663-693.
(pdf)
- (with B. Riviere) Convergence of a Discontinuous Galerkin Method
for the Miscible Displacement Equations Under Minimal Regularity,
SIAM J. Numer. Anal. Vol. 49, No. 3, pp 1085--1110, 2011.
(pdf)
- (with A. Rand) Delaunay Refinement Algorithms for Estimating
Local Feature Size in 2D and 3D,
Internat. J. Comput. Geom. Appl. 21 (2011), no. 5, pp 507-643.
(pdf)
- Compactness Properties of the DG and CG Time Stepping Schemes
for Parabolic Equations,
SIAM J. Numer. Anal. Vol. 47, No. 6, pp. 4680-4710, Feb 2010.
(pdf)
- (with K. Chrysafinos) Discontinuous Galerkin approximations of
the Stokes and Navier-Stokes equations,
Math. Comp. 79, No. 272, October 2010, pp. 2135-2167.
(pdf)
- (with K. Chrysafinos) Lagrangian and moving mesh methods
for the convection diffusion equation,
M2AN Math. Model. Numer. Anal. 42 (2008), no. 1, pp 25--55.
(pdf)
- (with C. Liu) Convergence of Numerical Approximations of the
Incompressible Navier-Stokes Equations with Variable Density and
Viscosity,
SINUM, Vol. 45, No. 3, pp 1287--1204, 2007.
(pdf)
- Convergence of the Discontinuous Galerkin Method for
Discontinuous Solutions,
SINUM, Vol. 44, No. 1, 2006, pp 349--366.
(pdf)
- (with G. Miller and S. Pav)
When and Why Delaunay Refinement Algortihms Work
IJCGA, Vol. 15, No. 1, 2005, pp 25--54.
(pdf)
- (with C. Liu)
An Eulerian Description of Fluids Containing Visco-elastic
Particles
Arch. Rational Mech. Anal. 159, 2001, pp 229--52.
(pdf)