Minisymposium ''Perspectives of Gradient-Robustness''
Berlin, July 08, 2021

In recent years many advanced finite element methods with improved robustness properties have been proposed. We are thinking of mixed and discontinuous Galerkin methods, with partial continuity of vector fields, and facet variables for hybridization.
In particular H(div) - based methods lead to pressure robustness for flow problems and nearly incompressible materials. This can be obtained by using H(div) - conforming methods from the beginning, or by updating classical methods using Alexander's reconstruction operators.
We give an overview of such methods, and show how to formulate them in within the language of the finite element library NGSolve.
We present joint work with C. Lehrenfeld, P. Lederer, A. Linke, and C. Merdon.

Anisotropic finite element meshes are particularly efficient when the solution of the problem has anisotropic features like boundary layers or edge singularities. The Crouzeix-Raviart method is known to be inf-sup stable on anisotropic meshes, yet not pressure-robust. Inf-sup stable finite element schemes with discontinuous pressure can be made pressure-robust by a modified discretization of the exterior forcing term using H(div)-conforming reconstruction operators like the Raviart-Thomas or Brezzi-Douglas-Marini interpolants.
In order to show that the reconstruction approach works for anisotropic discretizations, it was necessary to investigate the interpolation error for the Raviart-Thomas or Brezzi-Douglas-Marini interpolants on anisotropic elements disclosing subtleties in the definition of the spaces and in shape assumptions.
In collaboration with Alexander Linke and Christian Merdon we have generalized the modified Crouzeix-Raviart method to a large class of anisotropic triangulations. Numerical examples confirm the theoretical results.

This talk discusses error estimates of the pressure-robust schemes for the Stokes problem introduced by A. Linke and collaborators. These methods introduce a nonstandard discretization of the right-hand side function, leading to consistency errors. We discuss various approaches and mathematical tools to analyze the methods with a focus on low-regularity solutions. This is joint work with Alexander Linke, Christian Merdon, and Felix Neumann.

In high Reynolds number flows there are two possible sources of discretisation errors: a lack of gradient-robustness and dominant convection. The latter one persists if divergence-free discretisations like the Scott-Vogelius element are used and requires some kind of convection stabilisation. However, the gains from gradient-robustness can be compromised if a convection stablisation is used that is not gradient-robust like the SUPG convection stabilisation. The talk presents a novel residual-based least-square convection stabilisation which is based on the vorticity equation. This stabilisation preserves the gradient-robustness of the original method and so allows pressure-independent velocity estimates and the best known rates for the $L^2$-norm in the linearised Oseen case [1].
The second part focuses on compressible flows for low Mach numbers where gradient-robustness implies a well-balanced property in the sense that hydrostatic atmosphere-at-rest scenarios are preserved optimally. A novel finite element finite volume method is presented that is based on a gradient-robustly reconstructed version of the Bernardi--Raugel element and preserves total mass constraints, positiveness of the density, and is provably convergent on unstructured grids (at least in the simplest isothermal case) [2].
Both parts include some numerical examples to illustrate the findings. In the remaining time some future perspectives are discussed.

References:
[1] "A pressure-robust discretization of Oseen's equation using stabilization in the vorticity equation", N. Ahmed, G. R. Barrenechea, E. Burman, J. Guzman, A. Linke and C. Merdon, (accepted by SINUM, 2021)
[2] "A gradient-robust well-balanced scheme for the compressible isothermal Stokes problem", M. Akbas, T. Gallouet, A. Gassmann, A. Linke and C. Merdon, Computer Methods in Applied Mechanics and Engineering 367 (2020)