Discrete Exterior Calculus
Visualization of a vector-field resulting from a vector diffusion process on a horse-like manifold.
Discrete Exterior Calculus (DEC) is a grid-based geometric discretization technique for differential/integral forms and exterior derivatives. The basic idea is to associate geometric entities of a grid (like vertices, edges, or faces) with integral values, corresponding to integral of discrete differential forms over these entities. Then, derivatives are defined in a way that follows Stokes low of integration. For a flat hexagonal grid, a DEC discretization of a second order derivative, like a Laplacian, leads to the classical 5-point stencil in finite-differences context for 0-forms, or a stencil on the dual-grid for n-forms.
Research focus
This discretization technique is mainly used in computer graphics, because of its efficiency for some types of PDE problems. But, there is not much (numerical) analysis done and we want to close this gap with our research. Therefore, we analyze the consistency of the discretization for some differential operators depending on properties of the grid.
To be a usefull discretization method in scientific computing, efficient linear solvers for the resulting linear systems must be constructed. We want to analyze (geometric) multigrid solvers for this (geometric) discretization. Thereby, we combine ideas already developed in different fields into one formulation.
Implementing the DEC
For a toolbox that can handle the discrete exterior calculus (DEC) discretization, a grid must be equipped with some additional data. First of all, a DEC discretization needs geometric values, like volume and orientation, of alle entities of all dimension in the grid. Next, we need incidence and adjacency information between different entities. This allows to traverse neighbours and connected entities. A third information is an implicit representation of a dual grid.
We want to develop an efficient and user friendly C++ toolbox for a discretization based on DEC. It is based on the DUNE framework for a flexible handling of different grid implementations.
Student projects and thesis topics
- Analysis of an adaptive DEC discretization regarding consistency and convergence
- Edge based multigrid for a DEC discretization of the Stokes equation on curved manifolds
- Implementation of a 3d simplex-based grid adaptor that adds all necessary information to an existing grid implementation.
- Handling of hexagonal and cube grids
- Structured simplex grids including DEC discretization information for rectangular domains
- GPU-Parallelization of the DEC discretization
- Adaptivity based on domain decomposition techniques