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 (such as vertices, edges, or faces) with integral values corresponding to the integral of discrete differential forms over these entities. Then the derivatives are defined in a way that follows the Stokes low of the integration. For a flat hexagonal grid, a DEC discretization of a second-order derivative, such as a Laplacian, leads to the classical 5-point stencil in the 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. However, there is not much (numerical) analysis done and we want to fill this gap with our research. Therefore, we analyze the consistency of the discretization for some differential operators depending on the properties of the grid.
To be a useful discretization method in scientific computing, efficient linear solvers must be constructed for the resulting linear systems. We want to analyze (geometric) multigrid solvers for this (geometric) discretization. In doing so, we combine ideas already developed in different fields into one formulation.
Implementing the DEC
For a toolbox to handle Discrete Exterior Calculus (DEC) discretizations, a mesh must be provided with some additional data. First, a DEC discretization needs geometric values, such as volume and orientation, of all entities of all dimensions in the grid. Next, we need incidence and adjacency information between different entities. This allows us to traverse neighbors 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 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