Nonconforming finite elements for the Brinkman and -curlΔ curl problems on cubical meshes
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We propose two families of nonconforming elements on cubical meshes: one for the -curlΔcurl problem and the other for the Brinkman problem. The element for the -curlΔcurl problem is the first nonconforming element on cubical meshes. The element for the Brinkman problem can yield a uniformly stable finite element method with respect to the parameter ν. The lowest-order elements for the -curlΔcurl and the Brinkman problems have 48 and 30 degrees of freedom, respectively. The two families of elements are subspaces of H(curl;Ω) and H(div;Ω), and they, as nonconforming approximation to H(gradcurl;Ω) and [H^1(Ω)]^3, can form a discrete Stokes complex together with the Lagrange element and the L^2 element.
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