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A mass conserving mixed stress formulation for the Stokes equations. (English) Zbl 1466.65189

A new method for the variational mixed stress formulation of the Stokes equations is introduced, called mass conserving mixed formulation with stresses (MCS). The velocity \(u\) is approximated with \(H(\operatorname{div})\) conforming finite elements providing exact mass conservation. Exact mass conservation leads to a structure-preservation property called pressure robustness. New nonconforming finite elements are constructed for a new stress-like variable \(\sigma\) equalling the gradient of the velocity. It leads to a variational formulation requiring less regularity for \(u\) and a new function space for \(\sigma\). The stability in certain discrete norms and convergence of the new method are proved. Various numerical examples are presented to validate the theoretical results.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows
35Q35 PDEs in connection with fluid mechanics
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