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

Abstract: We develop a method to compute $H<sup>2$-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 $H<sup>1$-smoothness, so that conforming discretizations require at most $C<sup>0$-continuity. The method is demonstrated on arbitrary order $C<sup>1$-splines.