Decoupling Stokes
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We present new finite elements to discretize Stokes flow in two and three dimensions using high-order polynomials on both simplices and their Alfeld macro-refinement. These elements are designed for augmented Lagrangian preconditioners, and produce a stiffness matrix that decouples the interior bubbles and the high-order divergence-free bubbles from the interface basis functions. Although the decoupling is only achieved on the reference simplex, the mapping from the reference simplex to a physical cell introduces some coupling. We show that the coupling is weak and does not depend on the augmented Lagrangian penalty coefficient, and therefore can be ignored in a parallel subspace correction method. We obtain a robust and scalable solver by using lower-order elements as the coarse space. We illustrate our solver with numerical examples in Firedrake.