Fortin operator for the Taylor–Hood element

We design a local Fortin operator for the lowest-order Taylor–Hood element in any dimension, which was previously constructed only in 2D. In the construction we use tangential edge bubble functions for the divergence correcting operator. This naturally leads to an alternative inf-sup stable reduced finite element pair. Furthermore, we provide a counterexample to the inf-sup stability and hence to existence of a Fortin operator for the $$P_2$$ P 2 –$$P_0$$ P 0 and the augmented Taylor–Hood element in 3D.

Versatile stabilized finite element formulations for nearly and fully incompressible solid mechanics

Computational formulations for large strain, polyconvex, nearly incompressible elasticity have been extensively studied, but research on enhancing solution schemes that offer better tradeoffs between accuracy, robustness, and computational efficiency remains to be highly relevant. In this paper, we present two methods to overcome locking phenomena, one based on a displacement-pressure formulation using a stable finite element pairing with bubble functions, and another one using a simple pressure-projection stabilized $$\mathbb {P}_1 - \mathbb {P}_1$$P1-P1 finite element pair. A key advantage is the versatility of the proposed methods: with minor adjustments they are applicable to all kinds of finite elements and generalize easily to transient dynamics. The proposed methods are compared to and verified with standard benchmarks previously reported in the literature. Benchmark results demonstrate that both approaches provide a robust and computationally efficient way of simulating nearly and fully incompressible materials.

Crouzeix-Raviart MsFEM with Bubble Functions for Diffusion and Advection-Diffusion in Perforated Media

The adaptation of Crouzeix-Raviart finite element in the context of multi-scale finite element method (MsFEM) is studied and implemented on diffusion and advection-diffusion problems in perforated media. It is known that the approximation of boundary condition on coarse element edges when computing the multiscale basis functions critically influences the eventual accuracy of any MsFEM approaches. The weakly enforced continuity of Crouzeix-Raviart function space across element edges leads to a natural boundary condition for the multiscale basis functions which relaxes the sensitivity of our method to complex patterns of perforations. Another ingredient to our method is the application of bubble functions which is shown to be instrumental in maintaining high accuracy amid dense perforations. Additionally, the application of penalization method makes it possible to avoid complex unstructured domain and allows extensive use of simpler Cartesian meshes.

Stability and Convergence of Two Three-Field Finite Element Formulations for Elasticity

In this work analyses are carried out to verify the well-posedness of two three-field formulations for linear elasticity that have been shown to work well in practice, but for which analyses are absent. Both are formulated in three dimensions on tetrahedral elements. The first has as primary unknowns the displacement, pressure, and dilatation, and the second is a formulation with primary unknowns the displacement, pressure and enhanced strain, with the space of enhanced strains spanned by functions that are bubble functions on a surface of the element. It is shown that the first-mentioned formulation is a special case of a general three-field formulation, for which well-posedness conditions have been established. These conditions are used here to show that the pressure-dilatation-displacement formulation is uniformly convergent in the incompressible limit. The pressure-enhanced strain-displacement formulation is not amenable to such an approach; instead, it is reformulated in the form of a standard discrete saddle point problem, and the uniform convergence of the formulation established directly by verifying the ellipticity and inf-sup conditions.