This library contains an add-on to FEniCSx enabling 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 [5][8], elasticity [1][9] or poro-elasticity [2][10].

The equilibration process 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 [8]. In order to improve computational efficiency, a so called semi-explicit strategy [3][6] 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 - the stress tensor has a distinct symmetry - an additional constrained minimisation step after the row wise reconstruction of the tensor [1] is implemented.

[1] Bertrand, F., Kober, B., Moldenhauer, M. and Starke, G.: Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity. Comput. Math. Appl. (2021) doi: 10.1002/num.22741

[2] Bertrand, F. and Starke, G.: A posteriori error estimates by weakly symmetric stress reconstruction for the Biot problem. Numer. Methods Partial Differ. Equ. (2021) doi: 10.1016/j.camwa.2020.10.011

[3] Bertrand, F., Carstensen, C., Gräßle, B. and Tran, N.T.: Stabilization-free HHO a posteriori error control. Numer. Math. (2023) doi: 10.1007/s00211-023-01366-8

[4] Boffi, D., Brezzi, F. and Fortin, M.: Mixed finite element methods and applications. Springer Heidelberg, Berlin (2013).

[5] Braess, D. and Schöberl, J.: Equilibrated Residual Error Estimator for Edge Elements. Math. Comput. 77, 651-672 (2008)

[6] Cai, Z. and Zhang, S.: Robust equilibrated residual error estimator for diffusion problems: conforming elements. SIAM J. Numer. Anal. (2012). doi: 10.1137/100803857

[7] Kim, K.-Y.: Guaranteed A Posteriori Error Estimator for Mixed Finite Element Methods of Linear Elasticity with Weak Stress Symmetry. SIAM J. Numer. Anal. (2015) doi: 10.1137/110823031

[8] Ern, A and Vohralı́k, M.: Polynomial-Degree-Robust A Posteriori Estimates in a Unified Setting for Conforming, Nonconforming, Discontinuous Galerkin, and Mixed Discretizations. SIAM J. Numer. Anal. (2015) doi: 10.1137/130950100

[9] Prager, W. and Synge, J.L.: Approximations in elasticity based on the concept of function space. Q. J. Mech. Appl. Math. 5, 241-269 (1947)

[10] Riedlbeck, R., Di Pietro, D.A., Ern, A., Granet, S. and Kazymyrenko, K.: Stress and flux reconstruction in Biot’s poro-elasticity problem with application to a posteriori error analysis. Comput. Math. Appl. (2017) doi: 10.1016/j.camwa.2017.02.005

The installation of dolfinx_eqlb is described on github. The easiest way is the creation of a Docker container. Alternatively, a copy of a docker image containing the entire code, is provided in this data set. In order to use it, download the .tar.gz archive, navigate into the folder, where the download is located, and run

$ docker load --input dockerimage-dolfinx_eqlb-v1.2.0.tar.gz

The container can be started using

$ docker run -ti --rm brodbeck-m/dolfinx_eqlb:release

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