[1] O. Nemitz, M. B. Nielsen, M. Rumpf, and R. Whitaker. Narrow band methods for PDEs on very large implicit surfaces. In H. P. A. Lensch, B. Rosenhahn, H.-P. Seidel, P. Slusallek, and J. Weickert, editors, Vision, Modeling and Visualization, pages 171-180, 2007.
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Physical simulation on surfaces and various applications in geometry processing are based on partial differential equations on surfaces. The implicit representation of these eventually evolving surfaces in terms of level set methods leads to effective and flexible numerical tools. This paper addresses the computational problem of how to solve partial differential equations on level sets with an underlying very high-resolution discrete grid. These high-resolution grids are represented in a very efficient format, which stores only grid points in a thin narrow band. Reaction diffusion equations on a fixed surface and the evolution of a surface under curvature motion are considered as model problems. The proposed methods are based on a semi implicit finite element discretization directly on these thin narrow bands and allow for large time steps. To ensure this, suitable transparent boundary conditions are introduced on the boundary of the narrow band and the time discretization is based on a nested iteration scheme. Methods are provided to assemble finite element matrices and to apply matrix vector operators in a manner that do not incur additional overhead and give fast, cache-coherent access to very large data sets.