Two and three dimensional H²-conforming finite element approximations without C¹-elements

We develop a method to compute H²-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 H¹-smoothness, so that conforming discretizations require at most C⁰-continuity. The method is demonstrated on arbitrary order C¹-splines.