@article{car-cun-gom-sch-sto-06-dyad,
  author = {Cl{\'a}udio G. S. Cardoso and Maria Cristina C. Cunha and Anamaria Gomide and Denis J. Schiozer and Jorge Stolfi},
  title = {Finite Elements on Dyadic Grids with Applications},
  journal = {Mathematics and Computers in Simulation},
  volume = {73},
  number = {1--4},
  pages = {87--104},
  year = 2006,
  month = nov,
  publisher = {Elsevier},
  doi = {10.1016/j.matcom.2006.06.024},
  comment = {We have incorrectly cited this paper as ``to appear in Applied Numerical Mathematics (Elsevier).'' Apparently that was the original intention of the PANAM V organizers, but they choose MaCiS in the end. The title has also changed from ``Finite Elements on Dyadic Grids for Oil Reservoir Simulation'' (as originally submitted, and sometimes cited) to the one above, at the request of a MaCiS referee.},
  altkeys = {car-cun-gom-sch-sto-05-dyap},
  citations = {WoS/2021: 3},
  abstract = {A \emph{dyadic grid} is a $d$-dimensional hierarchical mesh where a cell at level $k$ is partitioned into two equal children at level $k+1$ by a hyperplane perpendicular to coordinate axis $(k \bmod d)$. We consider here the finite element approach on adaptive grids, static and dynamic, for various functional approximation problems. We review here the theory of adaptive dyadic grids and splines defined on them. Specifically, we consider the space $\CP_c^d[G]$ of all functions that, within any leaf cell of an arbitrary finite dyadic grid $G$, coincide with a multivariate polynomial of maximum degree $d$ in each coordinate, and are continuous to order $c$. We describe algorithms to construct a finite-element basis for such spaces. We illustrate the use of such basis for interpolation, least-squares approximation, and the Galerkin-style integration of partial differential equations, such as the heat diffusion equation and two-phase (oil/water) flow in porous media. Compared to tetrahedral meshes, the simple topology of dyadic grids is expected to compensate for their limitations, especially in problems with moving fronts.}
}