Finite Elements on Dyadic Grids with Applications Original Research Article

A dyadic grid is a d-dimensional hierarchical mesh where a cell at level k is partitioned into two equal children at level image by a hyperplane perpendicular to coordinate axis image. 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 image 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.