A Hellan-Herrmann-Johnson-like method for the stream function formulation of the Stokes equations in two and three space dimensions
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We introduce a new discretization for the stream function formulation of the incompressible Stokes equations in two and three space dimensions. The method is strongly related to the Hellan-Herrmann-Johnson method and is based on the recently discovered mass conserving mixed stress formulation [J. Gopalakrishnan, P.L. Lederer, J. Schöberl, IMA Journal of numerical Analysis, 2019] that approximates the velocity in an H(div)-conforming space and introduces a new stress-like variable for the approximation of the gradient of the velocity within the function space H(curldiv). The properties of the (discrete) de Rham complex allows to extend this method to a stream function formulation in two and three space dimensions. We present a detailed stability analysis in the continuous and the discrete setting where the stream function ψ and its approximation ψ_h are elements of H(curl) and the H(curl)-conforming Nédélec finite element space, respectively. We conclude with an error analysis revealing optimal convergence rates for the error of the discrete velocity u_h = curl(ψ_h) measured in a discrete H^1-norm. We present numerical examples to validate our findings and discuss structure-preserving properties such as pressure-robustness.
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