Simple and sharp: Error estimates of Bank–Weiser type in the FEniCS Project

We propose a simple, cheap and parallelisable implementation [1] in DOLFIN and DOLFINx of an implicit a posteriori error estimator introduced in [2].

The computation of the implicit estimator requires the solution of local Neumann problems in non-standard finite element spaces on each cell of the mesh. These special spaces are usually not available in modern automated finite element software, including the FEniCS Project.

Our method bypasses this issue by constructing a linear system on each cell corresponding to the problem in an available finite element space. We restrict this linear system to a non-standard space. On affine-equivalent finite elements, this restriction is constant and its application involves only small dense matrix-matrix multiplications.

We show several numerical examples of adaptive mesh refinement driven by this estimator applied to Poisson, Stokes and incompressible linear elasticity, as well as for goal-oriented problems [3].

References

• [1] Raphaël Bulle, Jack S. Hale, Alexei Lozinski, Stéphane P. A. Bordas, Franz Chouly. Hierarchical a posteriori error estimation of Bank–Weiser type in the FEniCS Project, submitted, 2020.
• [2] R. E. Bank and A. Weiser. Some a posteriori error estimators for elliptic partial differential equations, 1985. [DOI: 10.2307/2007953]
• [3] Roland Becker, Elodie Estecahandy, and David Trujillo. Weighted marking for goal-oriented adaptive finite element methods, 2011. [DOI: 10.1137/100794298]