The curl–curl conforming virtual element method for the quad-curl problem

In this paper, we present the [Formula: see text]-conforming virtual element (VE) method for the quad-curl problem in two dimensions. Based on the idea of de Rham complex, we first construct three families of [Formula: see text]-conforming VEs, of which the simplest one has only one degree of freedom associated to each vertex and each edge in the lowest-order case, respectively. An exact discrete complex is established between the [Formula: see text]-conforming and [Formula: see text]-conforming VEs. We rigorously prove the interpolation error estimates, the stability of discrete bilinear forms, the coercivity and inf–sup condition of the corresponding discrete formulation. We show that the conforming VEs have the optimal convergence. Some numerical examples are given to confirm the theoretical results.