Nicolas Robidoux's home page (in need of a serious update)

I maintain the

Mimetic discretizations bibliography homepage.

I was also one of the organizers of the NSF funded

Workshop on Mimetic Discretizations of Continuum Mechanics.

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Personal information

• Assistant professor at the Department of Mathematics and Computer Science of Laurentian University.
  
  
• Before that, I was a postdoctoral fellow with Bob Russell and Manfred Trummer at the Department of Mathematics of Simon Fraser University.
  
  
• Thesis advisor: Stanly Steinberg of the Mathematics and Statistics Department of the University of New Mexico-Albuquerque.
  
  
• Curriculum Vitae:
  Postscript file resume.ps (200K).
  PDF file resume.pdf (100K).
  Gziped Postscript file resume.ps.gz (100K).
  
  
• Mug shot (photographer: Jim Verner):
  JPG file (20K).
  Ego size JPG (60K).
  
  

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Refereed publications

• Computer algebra and interpolation: a lesson plan, Journal of Symbolic Computation (1997) 23, 551-576.
  BibTeX entry (1K).
  
  The following short and elementary proceedings article is recommended as an introduction to natural (mimetic) discretization methods for div-grad systems. My doctoral thesis (see below), which is more technical and comprehensive, is a continuation of this work.
• A new method of construction of adjoint gradients and divergences on logically rectangular smooth grids, 261-272 in Finite Volumes for Complex Applications, F. Benkhaldoun and R. Vilsmeier, editors (Hermes, Paris, 1996).
  Abstract.
  Abstract in French (minus the accents).
  Postscript file rouen.ps (200K).
  Gziped Postscript file rouen.ps.gz (70K).
  BibTeX entry (1K).
  
  
  With Adrian Kitson and Robert I. McLachlan:
  
  
• Skew-adjoint finite difference methods, accepted for publication in the New Zealand Journal of Mathematics.
  
  
  With Patrick M. Knupp:
  
  
• A framework for variational grid generation: conditioning the Jacobian matrix with matrix norms, Society for Industrial and Applied Mathematics Journal on Scientific Computing 21 (2000), no. 6, 2029-2047.
  Summary (Mathematical Reviews 2002a:65148).
  The article.
  BibTeX entry (1K).
  
  
  With Robert I. McLachlan:
  
  
• Antisymmetry, pseudospectral methods, and conservative PDEs, in International Conference on Differential Equations, Proceedings of the conference (Equadiff 99) held in Berlin, August 1-7, 1999, B. Fiedler, K. Groger and J. Sprekels, editors (World Scientific, Singapore, 2000), vol. 2, 994-999.
  MathScinet listing.
  Abstract.
  Postscript file skew.ps (200K).
  Gziped Postscript file skew.ps.gz (70K).
  BibTeX entry (1K).
  
  
  With Robert I. McLachlan and G. Reinout W. Quispel:
  
  
• Geometric integration using discrete gradients, Royal Society of London Philosophical Transactions, Series A, Mathematical, Physical and Engineering Sciences (1999) 357, no. 1754, 1021-1046.
  Steve Shkoller's review/summary (Mathematical Reviews 2000f:34021).
  BibTeX entry (1K).
  
  
• Unified approach to Hamiltonian systems, Poisson systems, gradient systems, and systems with Lyapunov functions or first integrals, Physical Review Letters (1998) 12, 2399-2403.
  Summary (Mathematical Reviews 99f:58079).
  The article.
  BibTeX entry (1K).
  
  
  

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Selected other publications

• My doctoral thesis Numerical solution of the steady diffusion equation with discontinuous coefficients. Describes special properties of 1D versions of current favorite discretizations of the divergence, the diffusion tensor, and the gradient. (155 pages.)
  Postscript file thesis.ps (1000K).
  Gziped Postscript file thesis.ps.gz (400K).
  PDF file thesis.pdf (500K).
  BibTeX entry (1K).
  
  
  With Hoon Hong, Richard Liska and Stanly Steinberg:
  
  
• Elimination of variables in parallel, SIAM News 33 (2000), no. 8, 1 and 12-13, in the column Applications on Advanced Architecture Computers, G. Astfalk, editor.
  The article
  
  
  

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Downloadable preprints

• Preprint of A discrete calculus in tensor grids by Stanly Steinberg and yours truly. This paper is meant to be fairly elementary. In fairly good shape although some proofs need to be redone. (59 pages.)
  Postscript file discrete_intro.ps (600K).
  Gziped Postscript file discrete_intro.ps.gz (200K).
  
  
• Preprint of Polynomial histopolation, superconvergent degrees of freedom, and pseudospectral discrete Hodge operators by yours truly. I like this paper a lot. The most important consequence of the material presented in this paper is that discrete Hodge stars from fluxes to circulations (based on face elements) and discrete Hodge star operators from masses to point values (based on histopolation) can be made superconvergent simultaneously when using hexahedral master elements. In need of a complete rewrite. (33 pages.)
  Postscript file histogram.ps (800K).
  Gziped Postscript file histogram.ps.gz (300K).
  PDF file histogram.pdf (500K).
  
  
• Preprint of Antisymmetry, pseudospectral methods, weighted residual discretizations, and energy conserving partial differential equations by Robert McLachlan and yours truly.
  Covers a lot of ground. The link to mixed Finite Element Methods needs to be made explicit, and the writing needs to be tightened up (there are some minor inconsistencies in the exposition). Read the above Antisymmetry, pseudospectral methods, and conservative PDEs first. (37 pages.)
  Abstract.
  Postscript file skew.ps (2000K).
  Gziped Postscript file skew.ps.gz (300K).
  
  
• Draft of my first pass at a doctoral thesis as it stood in 1997 (time of my ABD flight to the southern hemisphere) Natural Finite Volume Discretizations of the Gradient, Divergence, Laplacian, Diffusion Tensor and Advective Term on Grids with Quadrilateral Cells.
  Lots of numerical test results. The schemes described and tested here coincide in 1D with the ones analysed in the final thesis when the diffusion coefficient is constant on each cell. (83 pages.)
  Abstract.
  Postscript file oldthes.ps (600K).
  Gziped Postscript file oldthes.ps.gz (200K).
  Gziped Postscript file oldthes.bib (1K).
  BibTeX entry (1K).
  
  
  

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Current research interests

• Numerical analysis:
  
  Mixed methods based on the generalized Stokes Theorem and discrete constitutive relations. Discrete Hodge star operators. Superconvergent face elements. Numerical homogenization.
  
  Discrete Hodge-Helmoltz decomposition. Numerical solution of div-grad and curl-curl systems. Numerical computation of eddy currents.
  
  Conservative and structure preserving finite volume difference discretizations of the diffusion operator with rough heterogeneous, non-isotropic media (discontinuous degenerate full tensor diffusion coefficients, unbounded and discontinuous source terms, non-smooth grids with hexahedral cells).
  
  Numerical solution of nonlinear parabolic equations. Block pseudospectral solution of the diffusion equation.
  
  Structural factorizations of numerical differential operators.
  
  Conservative and structure preserving weighted residual, spectral, pseudospectral and finite element discretizations of (nonlinear) Hamiltonian (Lie-Poisson) partial differential equations.
  
  
• Grid generation:
  
  Application of differential geometry to the variational generation of structured grids (functionals involving the Jacobian matrix, metric tensor, or Hessian of the induced mapping).
  
  
• Mathematical physics:
  
  Existence of smooth factorizations of differential operators inducing single or multiple conserved quantities and/or Lyapunov functionals.
  

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Last major modification January 16, 2004.

Comments/inquiries: nrobidoux@cs.laurentian.ca (Nicolas Robidoux)