dolfinx for flux equilibration (dolfinx_eqlb)

Description

dolfinx_eqlb is an open source library, extending FEniCSx by local flux equilibration strategies. The resulting H(div) conforming fluxes can be used for the construction of adaptive finite element solvers for the Poisson problem [1], elasticity [2][3][4] or poro-elasticity [5][6].

The flux equilibration relies on so called patches, groups of all cells, connected with one node of the mesh. On each patch a constrained minimisation problem is solved [7]. In order to improve computational efficiency, a so called semi-explicit strategy [8][9] is also implemented. The solution procedure is thereby split into two steps: An explicit determination of an H(div) function, fulfilling the minimisation constraints, followed by an unconstrained minimisation on a reduced, patch-wise ansatz space. If equilibration is applied to elasticity -- stress tensors have distinct symmetry properties -- an additional constrained minimisation step, after the row wise reconstruction of the tensor [3][4] is implemented.

Features

dolfinx_eqlb supports flux equilibration on two-dimensional domains with arbitrary triangular grids. It further includes the following features

• A local projector into arbitrary function-spaces with cell-wise support
• A hierarchic Raviart-Thomas element based on Boffi, Brezzi and Fortin [10]
• Boundary conditions for H(div) spaces on general boundaries
• Flux equilibration based on Ern and Vohralik (FluxEqlbEV) or a semi-explicit strategy (FluxEqlbSE)
• Stress equilibration considering distinct symmetry properties in a weak sense

Getting started

1. Clone this repository using the command:

git clone git@github.com:brodbeck-m/dolfinx_eqlb.git

2. Download the required Docker image of DOLFINx:

docker pull dolfinx/dolfinx:v0.6.0-r1

3. Build a Docker image containing dolfinx_eqlb

cd docker
./build_image.sh 

4. Try out the basic demos. If no errors are reported, the equilibration process works as expected.

./launch-container.sh

# Equilibration for a Poisson problem
cd ./root/dolfinx_eqlb/python/demo/poisson
python3 demo_reconstruction.py  

# Equilibration for linear elasticity
cd ./root/dolfinx_eqlb/python/demo/elasticity
python3 demo_reconstruction.py  

Equilibrated fluxes for a-posteriori error estimation

Based on equilibrated fluxes reliable error estimates for different problem classes can be constructed. Showcases for the Poisson problem (estimate by Ern and Vohralik [2]) and linear elasticity (following Bertrand et al. [3]) are provided in the demo section:

./launch-container.sh

# Equilibration for a Poisson problem
cd ./root/dolfinx_eqlb/python/demo/poisson
python3 demo_error_estimation.py  

# Equilibration for linear elasticity
cd ./root/dolfinx_eqlb/python/demo/elasticity
python3 demo_error_estimation.py  

Therein, a manufactured solution is solved on a rectangular domain, discretised by a series of uniformly refined meshes. The actual error, contributions of the error estimate and convergence ratios are reported in a csv-file.

This approach can now be transferred to other problems.

Literature

[1] Braess, D. and Schöberl, J.: Equilibrated Residual Error Estimator for Edge Elements (2008).

[2] Prager, W. and Synge, J. L.: Approximations in elasticity based on the concept of function space (1947).

[3] Bertrand et al.: Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity (2021).

[4] Bertrand et al.: Weakly symmetric stress equilibration for hyperelastic material models (2020).

[5] Riedlbeck et al.: Stress and flux reconstruction in Biot’s poro-elasticity problem with application to a posteriori error analysis (2017).

[6] Bertrand, F. and Starke, G.: A posteriori error estimates by weakly symmetric stress reconstruction for the Biot problem (2021).

[7] Ern, A. and Vohralik, M.: Polynomial-Degree-Robust A Posteriori Estimates in a Unified Setting for Conforming, Nonconforming, Discontinuous Galerkin, and Mixed Discretizations (2015).

[8] Cai, Z. and Zhang, S.: Robust equilibrated residual error estimator for diffusion problems: conforming elements (2012).

[9] Bertrand et al.: Stabilization-free HHO a posteriori error control (2023).

[10] Boffi et al.: Mixed finite element methods and applications (2013).

License

dolfinx_eqlb 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.

dolfinx_eqlb 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 dolfinx_eqlb. If not, see https://www.gnu.org/licenses/.