Gallery

Here are some results obtained with the DUNE-FEM discretization module.

Higher order and adaptive Lagrange spaces

The first picture shows the solution to the Laplace equation on a locally adapted grid (100.000 triangles) using first order polynomials. The second picture shows the results for the same problem but using four cube elements and polynomial degree 6. Both have about the same discretization error.

Partial differential equations on surfaces

Heat equation on a moving surface

We solve the heat equation on a moving surface using a first order Lagrange space. The color show the solution which is initially constant and then increases in the regions where the surface is compressed.

Mean curvature flow

The next is mean curvature flow of a surface. The colors represent the distance of the surface from the origin.

Discontinous Galerkin Methods for Convection Dominated Problems

Here we consider a stabilized Discontinuous Galerkin (DG) method for hyperbolic and convection dominated problems. The presented scheme can be used in several space dimensions and with a wide range of grid element types for locally adaptive and load balanced parallel computations. More details about the scheme can be found here.
First we show a 2d computation of a forward facing step using YaspGrid (left) and ALUGrid (right):

The following is the same code using ALUGrid with local adaptivity on 64 processors:

The last example shows the solution to the same problem but using a finite-volume and a special filtered grid view included in DUNE-FEM - part of the grid is removed by prescribing a filter excluding the element inside a half sphere:

This work was partially funded by the Landesstiftung Baden-Würtemberg..

Discontinuous Galerkin methods for advection-diffusion equations

The following movie shows a cold bubble set in a stratified atmosphere. It falls down and slides along the ground, creating Kelvin-Helmholtz vortices. A 5th order compact discontinuous Galerkin 2 (CDG2) method with a 3rd order explicit Runge-Kutta scheme is used to solve the compressible Navier-Stokes equations.

This is part of a DFG funded project. More details about the implementation together with further atmospheric test cases can be found here. The page also includes some comparison of the DUNE-FEM code with the production code developed at the German weather service.

Medical image registration

Two surfaces are matched by some minimization procedure. Using preregistered images damaged parts can be reconstructed. The resulting model is solved by gradient flow, resulting in a parabolic equation with a non-local term. We used an adaptive DG method of second order.
In the following pictures blue is the target image, black is the image to be registered, and red is the resulting registration The first picture show the results without using preregistered images and the second picture is with the statistical model included.

The following shows the registration of a femur bone. Again a damaged specimen is being registered once without and once with statistical information.


In the last example the method is used to transform a marking (coloring of the mandible) from the reference image to a newly registered image.

In cooperation with the group of Thomas Vetter at the University of Basel.

Solution to the Shallow Water Equations

Discontinuous Galerkin method for 3D shallow water model

A 3D version of the shallow water model is solved. The time dependent domain is transformed to a fixed domain and the prismatic grid is used to facilitate the integration over water columns. The colors represent the magnitude of the horizontal velocity vector.

More details about the implementation and the settings used can be found here. This was part of a diploma thesis at the University of Freiburg.

Discontinuous Galerkin method for 2D shallow water model

The following movie shows a wave propagating through a realistic river bed. A 2D shallow water model is solved, taking into account wetting and drying and bottom friction.



Work funded by the BMBF project.

Finite-Volume scheme on surfaces

The shallow water equations is solved on the sphere including bottom topography. The movie shows the first 15 hours of the propagation of tsunami waves caused by an asteroid impact in the middle of the Atlantic Ocean. The overall time maximum of the wave height above sea level is colored, where red depicts a wave height of at least 10m, and blue depicts 0m.

The simulation was performed on an triangular surface grid with more than 3 million elements in a parallel run on 16 cores. The picture shows the partitioning of the sphere. More details about the implementation and the settings used can be found here.
This was part of a diploma thesis at the University of Freiburg.