Nonconforming finite element Stokes complexes in three dimensions
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Two nonconforming finite element Stokes complexes ended with the nonconforming P_1-P_0 element for the Stokes equation in three dimensions are constructed. And commutative diagrams are also shown by combining nonconforming finite element Stokes complexes and interpolation operators. The lower order H(grad curl)-nonconforming finite element only has 14 degrees of freedom, whose basis functions are explicitly given in terms of the barycentric coordinates. The H(grad curl)-nonconforming elements are applied to solve the quad-curl problem, and optimal convergence is derived. By the nonconforming finite element Stokes complexes, the mixed finite element methods of the quad-curl problem is decoupled into two mixed methods of the Maxwell equation and the nonconforming P_1-P_0 element method for the Stokes equation, based on which a fast solver is developed.
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