O. Nemitz, M. B. Nielsen, M. Rumpf, and R. Whitaker.
Finite element methods on very large, dynamic tubular grid encoded
SIAM Journal on Scientific Computing, 31(3):2258-2281, 2009.
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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.