Skip to article frontmatterSkip to article content
Simula Research Laboratory

Equilibration-based a-posteriori error estimators within the FEniCSx finite element framework

Authors
Affiliations
University of Stuttgart
University of Technology Chemnitz
University of Stuttgart

Numerical solutions of, e.g. poro-elasticity or heat-conduction suffer from the deterioration of the overall accuracy when re-entrant corners, internal respective boundary layers or shock-like fronts are present (Verführth, 1994). A-posteriori error estimates offers a systematic way of retaining accuracy by localised mesh refinements. Following the seminal idea of Prager and Synge (1947), such error estimates can be constructed based on the comparison of discontinuous fluxes, calculated from the approximation, and any H(div) conforming function, satisfying the equilibrium condition. This function is typically called the equilibrated flux.

In order to retain computational efficiency, the equilibrated flux is calculated locally. Focusing on conforming finite-element-discretisation, this requires the solution of constrained minimization problems on local patches (Ern & Vohralik, 2015). The computational efficiency can be increased by splitting the equilibration into an explicit step followed by an unconstrained minimisation on a modified H(div) space with patch-wise support (Bertrand et al., 2023).

Within this contribution, we discuss efficient implementations of different equilibration procedures within the finite element framework FEniCSx. Special attention will be paid to requirements resulting from the application of equilibration based error estimation in poro-elasticity.

References

Verfürth, R. (1994). A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math., 50, p. 67-83. doi:10.1016/0377-0427(94)90290-9

Prager, W. & Synge, J. L. (1947). Approximations in elasticity based on the concept of function space. Quart. Appl. Math., 5, p. 241–269. Retrieved from https://www.jstor.org/stable/43633616

Ern, A. & Vohralik, M. (2015). Polynomial-Degree-Robust A Posteriori Estimates in a Unified Setting for Conforming, Nonconforming, Discontinuous Galerkin, and Mixed Discretizations. SIAM J. Numer. Anal., 53 , p. 1058-1081. doi:10.1137/130950100

Bertrand, F., Carstensen, C., Gräßle, B. & Tran, N. T. (2023). Stabilization-free HHO a posteriori error control. Numer. Math., 154, p. 369–408. doi:10.1007/s00211-023-01366-8

Abstracts
Enabling unstructured-mesh computation on massively tiled AI processors: An example of accelerating in silico cardiac simulation
Abstracts
Estimation of optimal inlet boundary condition for blood flow assessment in abdominal aortic aneurysm using variational data assimilation