Narrow Band Methods for PDEs on Very Large Implicit Surfaces Oliver Nemitz 1 , Michael Bang Nielsen 2 , Martin Rumpf 3 , Ross Whitaker 4 1,3 University of Bonn, Germany, 2 University of ˚ Arhus, Denmark, 4 University of Utah, U.S.A. Email: 1,3 {oliver.nemitz,martin.rumpf}@ins.uni-bonn.de, 2 bang@daimi.au.dk, 4 whitaker@cs.utah.edu Abstract Physical simulation on surfaces and various appli- cations in geometry processing are based on par- tial 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 dif- ferential equations on level sets with an underly- ing very high-resolution discrete grid. These high- resolution grids are represented in a very efficient format, which stores only grid points in a thin nar- row band. Reaction diffusion equations on a fixed surface and the evolution of a surface under cur- vature 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 en- sure this, suitable transparent boundary conditions are introduced on the boundary of the narrow band and the time discretization is based on a nested it- eration 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. 1 Introduction This paper addresses the computational problem of how to solve partial differential equations (PDEs) on the level sets of smooth scalar functions that are approximated by very high-resolution discrete grids. The context for this work is the growing in- terest in computing PDEs on surfaces that are rep- resented implicitly as the level sets of a smooth scalar function φ. Starting with the pioneering pa- per by Osher and Sethian [31] this way has become increasing important in a variety of fields such as computational physics [3, 4, 6, 7, 20], scientific visu- alization [23], image analysis [5, 8], and computer graphics [27, 30]. Most of these applications rely on the efficient computation of partial differential equations on curves or surfaces implicitly repre- sented by a level set function φ resolved on a dis- crete, usually structured, grid. The attraction of solving problems with discretely sampled implicit surfaces is the relatively large number of degrees of freedom provided by the grid and the freedom of not having to choose an explicit surface parameter- ization, which often limits shape and topology. There are in particular two scenarios in which such surface-based PDEs are interesting. The first is when the implicit surface serves as the domain and one would like to solve a PDE for a function u intrinsic on the surface. Projections of the deriva- tives in the ambient space onto the surface pro- vide a mechanism for computing differential oper- ators that live on the surface [5]. The other sce- nario is when the surface itself evolves according to a geometric PDE that depends on the shape. The most prominent example is motion by mean cur- vature [16]. For the discretization in space either finite difference [31, 33] or finite element schemes [10] are considered. Semi-implicit time discretiza- tions are suitable due to their stability properties also for large time steps, compared to explicit time discretization for diffusion type problems which re- quire time steps of the grid size squared. This is par- ticularly important when one is considering higher order PDEs [14, 20]. Perhaps the greatest promise of level-set meth- ods, for both moving interfaces and PDEs defined on static surfaces (codimension one), is their ability to deal with a wide variety of complicated shapes in an elegant manner within a single computational framework. However, the computation and memory requirements on the discrete grid that represents φ become prohibitive as the grid resolution increases. VMV 2007 H. P. A. Lensch, B. Rosenhahn, H.-P. Seidel, P. Slusallek, J. Weickert (Editors)