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

TL;DR

This paper introduces a new finite element method for approximating $H^2$-conforming solutions in 2D and 3D that avoids the need for $C^1$-elements by using a mixed variational formulation with only $C^0$-continuity.

Contribution

It presents a novel mixed variational formulation enabling $H^2$-conforming finite element approximations using standard finite element spaces with at most $C^0$-continuity.

Findings

Method successfully applied to arbitrary order $C^1$-splines.

Achieves $H^2$-conformity without $C^1$-elements.

Applicable in both two and three dimensions.

Abstract

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.