Finite elements on dyadic grids with applications

  • Authors:

  • Cláudio G. S. Cardoso;Maria Cristina Cunha;Anamaria Gomide;Denis J. Schiozer;Jorge Stolfi

  • Affiliations:

  • Institute of Computing, University of Campinas, Brazil;Institute of Mathematics, Statistics, and Scientific Computing, University of Campinas, Brazil;Institute of Computing, University of Campinas, Brazil;School of Mechanical Engineering, University of Campinas, Brazil;Institute of Computing, University of Campinas, Brazil and IC-UNICAMP, Campinus, SP, Brazil

  • Venue:
  • Mathematics and Computers in Simulation - Special issue: Applied and computational mathematics - selected papers of the fifth PanAmerican workshop - June 21-25, 2004, Tegucigalpa, Honduras
  • Year:
  • 2006

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Abstract

A 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 mod 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 Pcd[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.