High-order and sparsity-promoting Stokes elements

One of the long-standing challenges of numerical analysis is the efficient and stable solution of incompressible flow problems (e.g. Stokes). It is fairly non-trivial to design a discretization that yields a well-posed (invertible) linear saddle-point problem. Additionally requiring that the discrete solution preserves the divergence-free constraint introduces further difficulty. In this talk, we present new finite elements for incompressible flow using high-order piecewise polynomials spaces. These elements exploit certain orthogonality relations to reduce the computational cost and storage of augmented Lagrangian preconditioners. We achieve a robust and scalable solver by combining this high-order element with a domain decomposition method, and a lower-order element as the coarse space. We illustrate our solver with numerical examples in Firedrake.