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 (2024-10-15)
