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.

- (with B. Sequin)
*Multi-component Multiphase Flow,*Submitted Archive for Rational Mechancics and Analysis, March 2017. (pdf) - (with L. Perrotti and D. Wang)
*Numerical Approximation of Viscoelastic Fluids,*M2AN Math. Model. Numer. Anal. 51 (2017), pp 1119-1144. (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)