In this paper we shall consider nonconforming finite element discretization of the following Stokes complex in three dimensions
where . Conforming finite element Stokes complexes on triangles and rectangles in two dimensions are devised in [22, 33]. And a conforming virtual element discretization of the Stokes complex (1.1) in three dimensions is advanced in . To the best of our knowledge, there is no finite element discretization of the Stokes complex (1.1) in three dimensions in literature. Recently -conforming finite elements in three dimensions are constructed with in , whose space of shape functions includes all the polynomials with degree no more than . The number of the degrees of freedom for the lowest-order element in  is . Due to the large dimension of the conforming element spaces, nonconforming elements to discretize in three dimensions are preferred. The -nonconforming Zheng-Hu-Xu element in  only has degrees of freedom.
We will construct an -nonconforming finite element possessing fewer degrees of freedom than those of the Zheng-Hu-Xu element, but preserving the same approximation error in energy norm. Then we build up the nonconforming finite element Stokes complexes from the new -nonconforming finite element and the Zheng-Hu-Xu element. The finite element discretization of in the Stokes complex (1.1) should be a stable divergence-free pair for the Stokes equation, which suggests us to use the nonconforming linear element and piecewise constant to discretize and respectively. On the other hand, the direct sum decomposition [1, 2] implies the curl operator is injective. This motivates us to take the space of shape functions with . Note that is exactly the space of shape functions of the Zheng-Hu-Xu element. The dimension of is , which is six fewer than the dimension of . The degrees of freedom for are given by
By comparing the degrees of freedom, the lower order nonconforming element for is very similar as the Morley-Wang-Xu element  for . The explicit expressions of the basis functions of are shown in terms of the barycentric coordinates. Furthermore, we construct the commutative diagram
The -nonconforming element together with the Lagrange element is applied to solve the quad-curl problem. The discrete Poincaré inequality is established for the -nonconforming element space , as a result the coercivity on the weak divergence-free space follows. Then we acquire the discrete stability of the bilinear form from the evident discrete inf-sup condition, and derive the optimal convergence of the nonconforming mixed finite methods. Since the interpolation operator is not well-defined on , in the error analysis we exploit a quasi-interpolation operator defined on , which is constructed by combining a regular decomposition for the space , and the Scott-Zhang interpolation operator .
By the nonconforming finite element Stokes complex in the bottom line of the commutative diagram (1.2), we equivalently decouple the mixed finite element methods of the quad-curl problem into two mixed methods of the Maxwell equation and the nonconforming - element method for the Stokes equation, as the decoupling of the quad-curl problem in the continuous level [9, 36]. A fast solver based on this equivalent decoupling is developed for the mixed finite element methods of the quad-curl problem.
in two dimensions, and
in three dimensions, where . We refer to [16, 26, 3, 25, 17, 19, 20, 35, 18] for some finite element discretizations of the Stokes complex (1.3) in two dimensions, and [31, 19, 29] for some finite element discretizations of the Stokes complex (1.4) in three dimensions. While the finite elements corresponding to the Stokes complexes (1.3)-(1.4) are not suitable to discretize the quad-curl problem, since is not a subspace of .
The rest of this paper is organized as follows. In Section 2, we devise a lower order -nonconforming finite element. Nonconforming finite element Stokes complexes are developed in Section 3. In Section 4, we propose the nonconforming mixed finite element methods for the quad- problem. And the decoupling of the mixed finite element methods and a fast solver are discussed in Section 5.
2. The -nonconforming finite elements
In this section we will present -nonconforming finite elements.
Given a bounded domain and a nonnegative integer , let be the usual Sobolev space of functions on , and
the vector version of. The corresponding norm and semi-norm are denoted, respectively, by and . Let be the standard inner product on or . If is , we abbreviate , and by , and , respectively. Denote by the closure of with respect to the norm . Let stand for the set of all polynomials in with the total degree no more than , and be the vector version of . Let be the -orthogonal projector, and its vector version is denoted by . Set . The gradient operator, curl operator and divergence operator are denoted by , and respectively. And define Sobolev spaces , , , and in the standard way.
Assume is a contractible polyhedron. Let be a regular family of tetrahedral meshes of . For each element , denote by the unit outward normal vector to , which will be abbreviated as for simplicity. Let , , and be the union of all faces, interior faces, edges and vertices of the partition , respectively. We fix a unit normal vector for each face , and a unit tangent vector for each edge . For any , denote by , and the set of all faces, edges and vertices of , respectively. For any , let be the set of all edges of . And for each , denote by the unit vector being parallel to and outward normal to . Set , where is the exterior product. For elementwise smooth function , define
Let , and be the elementwise version of , and with respect to .
2.2. Nonconforming finite elements
The decomposition (2.1) implies that is injective. We intend to use the nonconforming linear element to discretize , then the decomposition (2.1) and the complex (1.1) motivate us that the space of shape functions to discrete should include . The direct sum in (2.1) also suggests to enrich with for some positive integer to get the space of shape functions. Hence for each , define the space of shape functions as
By the decomposition (2.1), we have , and
Then choose the following local degrees of freedom
The degrees of freedom (2.2)-(2.3) are inspired by the degrees of freedom of nonconforming linear element and the Nédélec element [27, 28]. Note that the triple is exactly the nonconforming finite element in . Here we embed this nonconforming finite element into the discrete Stokes complex. And we also construct the lowest order triple .
For each , applying the integration by parts on face , we get from the vanishing degrees of freedom (2.2) that
which together with the vanishing degrees of freedom (2.3) implies
Since , we acquire from the unisolvence of the nonconforming linear element that . Thus there exists such that . By the vanishing degrees of freedom (2.2), it holds , which implies that we can choose such that for each . Noting that , we acquire and . ∎
By comparing the degrees of freedom, the lower order nonconforming element for is very similar as the Morley-Wang-Xu element  for .
2.3. Basis functions
We will figure out the basis functions of in the subsection. We refer to  for the basis functions of . Let , , and be the barycentric coordinates of point with respect to the vertices and of the tetrahedron respectively. Let be the face of opposite to . And the vertices of denoted by , and with . Set , which is a tangential vector to the edge with vertices and , and similarly define other tangential vectors with different subscripts. For ease of presentation, let
The degrees of freedom and are equivalent to , i.e. (2.3).
2.3.1. Basis functions corresponding to the face degrees of freedom
where constant . We will show that and are the basis functions being dual to and .
Since is constant, we have
Hence it follows
Therefore the identity (2.4) follows from the fact is parallel to . ∎
Functions and are the basis functions of being dual to and , respecrtively. That is
Apparently . Since , , , and is constant on , we get
Again, from , and (2.4), it follows
Similarly we can show that for other edges and . Hence (2.5) holds.
2.3.2. Basis functions corresponding to the edge degrees of freedom
Next we construct the basis function corresponding to the degree of freedom . Recall the basis function of the lowest order Nédélec element of the first kind . Thanks to (2.5)-(2.6), function can be modified by and to derive the basis function of corresponding to the degree of freedom .
for each and .
The identities and follow from (2.5) and the fact
On the other hand, we get from that
for . ∎
In summary, we arrive at the basis functions being dual to the degrees of freedom , and :
Two basis functions on each face ()
where constant .
One basis function on each edge ()
3. Nonconforming finite element Stokes complexes
where . Since , it holds , where
We can use the Lagrange element, the nonconforming linear element and piecewise constant to discretize , and in the Stokes complex (1.1), respectively. Take the Lagrange element space
with , the nonconforming linear element space
and the piecewise constant space
Here is the jump of across . Define the global -nonconforming element space
According to the proof of Lemma 2.1, it holds
where with . The degrees of freedom for are
The notation means the kernel space of the operator .
Apparently we have