FEniCS Error Estimation (FEniCS-EE)

Description

FEniCS-EE is an open source library showing how various error estimation strategies can be implemented in the FEniCS Project finite element solver (https://fenicsproject.org). A particular focus is on implicit hierarchical a posteriori error estimators, that usually involve solving local error problems in special finite element spaces on cells of the mesh.

FEniCS-EE is described in the pre-print:

Hierarchical a posteriori error estimation of Bank-Weiser type in the FEniCS Project, R. Bulle, J. S. Hale, A. Lozinski, S. P. A. Bordas, F. Chouly, (https://arxiv.org/abs/2102.04360).

FEniCS-EE is compatible with the 2019.2.0 development version of the FEniCS Project.

An experimental version for DOLFINX master is available on the branch jhale/dolfinx.

Features

FEniCS-EE currently includes implementations of the following error estimation techniques for the Poisson problem:

• Implicit residual estimator of Bank and Weiser,
• Implicit residual estimator of Verfürth,

the following error estimation techniques for the incompressible elasticity problem:

• Implicit residual estimator of Khan, Powell and Silvester (https://arxiv.org/abs/1710.03328),

and the following error estimation techniques for the Stokes problem:

• Implicit residual estimator of Liao and Silvester (https://doi.org/10.1016/j.apnum.2010.05.003).

The following marking strategies:

• Maximum (bulk),
• Dörfler (equilibration).

Upcoming features

Please see the issue tracker for proposed features.

Getting started

1. Install FEniCS by following the instructions at http://fenicsproject.org/download. We recommend using Docker to install FEniCS. However, you can use any method you want to install FEniCS.

2. Then, clone this repository using the command:

   git clone https://github.org/rbulle/fenics-error-estimation
   

3. If you do not have an appropiate version of FEniCS already installed, use a Docker container (skip the second line if you have already an appropiate version of FEniCS installed):

   ./launch-container.sh
   

4. You should now have a shell inside a container with FEniCS installed. Try out an example:

   python3 setup.py develop --user
   cd demo/pure_dirichlet
   python3 demo_pure-dirichlet.py
   

   The resulting fields are written to the directory output/ which will be shared with the host machine. These files can be opened using Paraview.

Automated testing

We use github actions to perform automated testing. All documented demos include basic sanity checks on the results. Tests are run in the rbulle/fenics-error-estimation Docker image.

FAQ

Question: Is this a replacement of, or a competitor with, the automated error estimation strategy already implemented in DOLFIN?

Answer: No, the examples in this repository are aimed at users who wish to implement their own a posteriori error estimation strategies into DOLFIN and to have full control over the mathematical and numerical formulation.

Question: Can you tackle goal-oriented mesh adaptivity problems?

Answer: Yes, see the demo demo/goal_oriented_adaptivity/demo_goal-oriented-adaptivity.py. We use a weighted marking strategy, as opposed to a weighted residual strategy, to control the error in the goal functional. This avoids solving the dual/adjoint problem in a higher-order space, or, ad-hoc extrapolation procedures.

Question: Does it work for higher-order polynomial finite element spaces?

Answer: Yes, the Bank-Weiser and Verfürth methods work for higher order polynomial finite element spaces.

Question: What will happen when FEniCSX https://fenicsproject.org/fenics-project-roadmap-2019/ is released?

Answer: There is a prototype for DOLFINX available on the branch jhale/dolfinx.

Question: What about method x?

Answer: We'd be happy to work with you to add your error estimation methodology to this repository.

Citing

Please consider citing the FEniCS-EE paper and code if you find it useful.

@misc{bulle2021hierarchical,
    title={Hierarchical a posteriori error estimation of Bank-Weiser type in the FEniCS Project},
    author={Raphaël Bulle and Jack S. Hale and Alexei Lozinski and Stéphane P. A. Bordas and Franz Chouly},
    year={2021},
    eprint={2102.04360},
    archivePrefix={arXiv},
    primaryClass={math.NA}
}

@misc{bulle_fenics-ee_2019,
      title = {{FEniCS} {Error} {Estimation} {(FEniCS-EE)}},
      author = {Bulle, Raphaël, and Hale, Jack S.},
      month = jan,
      year = {2019},
      doi = {10.6084/m9.figshare.10732421},
      keywords = {FEniCS, finite element methods, error estimation},
}

along with the appropriate general FEniCS citations.

Issues and Support

Please use the issue tracker to report any issues.

Authors (alphabetical)

Raphaël Bulle, University of Luxembourg, Luxembourg.

Jack S. Hale, University of Luxembourg, Luxembourg.

License

FEniCS-EE is free software: you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details.

You should have received a copy of the GNU Lesser General Public License along with FEniCS-EE. If not, see http://www.gnu.org/licenses/.