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.
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.)
file skew.ps (2000K). Gziped
Postscript file skew.ps.gz (300K).
Mixed methods based on the generalized Stokes Theorem and discrete
constitutive relations. Discrete Hodge star
operators. Superconvergent face elements. Numerical
Discrete Hodge-Helmoltz decomposition. Numerical solution of div-grad
and curl-curl systems. Numerical computation of eddy
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.
Application of differential geometry to the variational generation of
structured grids (functionals involving the Jacobian matrix, metric
tensor, or Hessian of the induced mapping).
Existence of smooth factorizations of differential operators inducing
single or multiple conserved quantities and/or Lyapunov functionals.