Pressure-robustness in the context of optimal control

This paper studies the benefits of pressure-robust discretizations in the scope of optimal control of incompressible flows. Gradient forces that may appear in the data can have a negative impact on the accuracy of state and control and can only be correctly balanced if their L2-orthogonality onto discretely divergence-free test functions is restored. Perfectly orthogonal divergence-free discretizations or divergence-free reconstructions of these test functions do the trick and lead to much better analytic a priori estimates that are also validated in numerical examples

Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes

In recent years, a great deal of attention has been paid to discretizations of the incompressible Stokes equations that exactly preserve the incompressibility constraint. These are of substantial interest because these discretizations are pressure-robust; i.e., the error estimates for the velocity do not depend on the error in the pressure. Similar considerations arise in nearly incompressible linear elastic solids. Conforming discretizations with this property are now well understood in two dimensions but remain poorly understood in three dimensions. In this work, we state two conjectures on this subject. The first is that the Scott–Vogelius element pair is inf-sup stable on uniform meshes for velocity degree k≥4; the best result available in the literature is for k≥6. The second is that there exists a stable space decomposition of the kernel of the divergence for k≥5. We present numerical evidence supporting our conjectures

Nonlinear elasticity complex and a finite element diagram chase

In this paper, we present a nonlinear version of the linear elasticity (Calabi, Kr\"oner, Riemannian deformation) complex which encodes isometric embedding, metric, curvature and the Bianchi identity. We reformulate the rigidity theorem and a fundamental theorem of Riemannian geometry as the exactness of this complex. Then we generalize an algebraic approach for constructing finite elements for the Bernstein-Gelfand-Gelfand (BGG) complexes. In particular, we discuss the reduction of degrees of freedom with injective connecting maps in the BGG diagrams. We derive a strain complex in two space dimensions with a diagram chase.Comment: Manuscript prepared for proceedings of the INdAM conference "Approximation Theory and Numerical Analysis meet Algebra, Geometry, Topology'', which was held in September 2022 at Cortona, Ital