author = {Nemitz, Oliver and Nielsen, Michael Bang and Rumpf, Martin and Whitaker,
  title = {Finite Element Methods on Very Large, Dynamic Tubular Grid Encoded
	Implicit Surfaces},
  journal = {SIAM Journal on Scientific Computing},
  year = {2009},
  volume = {31},
  pages = {2258--2281},
  number = {3},
  abstract = {The simulation of physical processes on interfaces and a variety of
	applications in geometry processing and geometric modeling are based
	on the solution of partial differential equations on curved and evolving
	surfaces. Frequently, an implicit level set type representation of
	these surfaces is the most effective and computationally advantageous
	approach. This paper addresses the computational problem of how to
	solve partial differential equations on highly resolved level sets
	with an underlying very high-resolution discrete grid. These high-resolution
	grids are represented in a very efficient Dynamic Tubular Grid encoding
	format for a narrow band. A reaction diffusion model on a fixed surface
	and surface evolution driven by a nonlinear geometric diffusion approach,
	by isotropic, or truly anisotropic curvature motion are investigated
	as characteristic model problems. The proposed methods are based
	on semi-implicit finite element discretizations directly on these
	narrow bands, require only standard numerical quadrature and allow
	for large time steps. To combine large time steps with a very thin
	and thus storage inexpensive narrow band, suitable transparent boundary
	conditions on the boundary of the narrow band and a nested iteration
	scheme in each time step are investigated. This nested iteration
	scheme enables the discrete interfaces to move in a single time step
	significantly beyond the domain of the narrow band of the previous
	time step. Furthermore, algorithmic tools are provided to assemble
	finite element matrices and to apply matrix vector operators via
	fast, cache-coherent access to the Dynamic Tubular Grid encoded data
	structure. The consistency of the presented approach is evaluated
	and various numerical examples show its application potential.},
  doi = {10.1137/080718334},
  pdf = { 1}