This code is designed to numerically solve the Poisson equation,
in 2D and 3D using the local discontinuous Galerkin (LDG) method from scratch.
The tutorial codes in the deal.ii library (step-12 and step-39) use the MeshWorker interface to build discontinuous Galerkin (DG) methods. While this is very convenient, I could not use this framework for solving my research roblem and I needed write the LDG method from scratch. I thought it would help for others to have access to this example that goes through writing a discontinuous Galerkin method from scatch and also shows how to do it in a distributed setting using the Trilinos library. Therefore, I submitted this code to the deal.II code-gallery, you can find it here. This example may also be of interest to users that wish to use the LDG method, as the method is distinctly different from the Interior Penalty Discontinuous Galerkin (IPDG) methods and was not covered in other tutorials on DG methods.
The LDG method is very useful when one is working with a differential equation and desires both approximations to the scalar unknown function as well as its flux. The mixed method a method where one can obtain both the scalar unknown function as we as its flux, however, the LDG method has less unknowns to solve for compared to the mixed finite element method when using the Raviart-Thomas element. The LDG method also approximates the scalar unknown function and its flux using discontinous polynomial basis function. This makes it much more suitable when one wishes to use local mesh refinement as hanging nodes are not an issue as they are in the mixed method or other finite element methods. For an introduction do the LDG method, see the write-up here.
The requirements for this software are,
- deal.ii library version 8.3.0 or higher,
- CMake version 2.8 or higher,
- MPI version 2/3.1.4,
- Trilinos version 11.12,
- p4est version 1.1,
- ParaView for visualization.
First obtain and install a copy of the dealii deal.ii library version 8.3.0 or higher as well as the other dependencies. See the dealii library for installation instructions and installation instructions for Trilinos and p4est.
To generate a makefile for this code using CMake, type the following command into the terminal from the main directory:
cmake . -DDEAL_II_DIR=/path_to_deal.ii
To compile the code in debug mode use:
make
To compile the code in release mode use:
make release
Either of these commands will create the executable, main, however the release mode will make a faster executable.
To run the code on N processor type the following command into ther terminal from the main directory,
mpirun -np N ./main
The output of the code will be .vtu and .pvtu and be written to disk in parallel. The results can be viewed using ParaView. An example of the results for a 3D problem are shown below,
