Computing \(H^2\)-conforming finite element approximations without having to implement \(C^1\)-elements (Q6585308)

The paper introduces a new method for the computation of \(H^2\)-conforming finite element approximations of the solutions of fourth-order elliptic equations on two-dimensional polygonal domains. The proposed approach eliminates the need for constructing \(C^1\)-elements and comprises three steps. First, an \(H^1\)-elliptic conforming projection is computed as a preprocessing step. Next, a Stokes-like problem is solved. Finally, another \(H^1\)-elliptic conforming projection is performed as a postprocessing step. Since all three steps require only \(H^1\)-conformity and utilize standard finite element schemes, the resulting algorithm can be efficiently implemented using existing software packages. Moreover, the method preserves both the conformity and stability of the original \(H^2\)-conforming formulation. The applicability of the method is demonstrated on several \(H^2\)-conforming elements, including Morgan-Scott, Argyris, and HCT elements. The theoretical findings are supported by numerical experiments, which include applications of the proposed method to Morgan-Scott elements for an L-shaped Kirchhoff plate with point load, a G-shaped plate with uniform load, and the numerical study of low-order elements.