Two and three dimensional $H^2$-conforming finite element approximations without $C^1$-elements

We develop a method to compute $H^{2$-conforming} finite element approximations in both two and three space dimensions using readily available finite element spaces. This is accomplished by deriving a novel, equivalent mixed variational formulation involving spaces with at most $H^{1$-smoothness,} so that conforming discretizations require at most $C^{0$-continuity.} The method is demonstrated on arbitrary order $C^1$-splines.