Two and three dimensional $H^2$-conforming finite element approximations without $C^1$-elements
Mark Ainsworth, Charles Parker
TL;DR
This paper tackles the challenge of obtaining $H^2$-conforming finite element approximations in 2D and 3D without $C^1$-elements by introducing a novel mixed variational formulation that uses only $H^1$-regular spaces. By introducing the auxiliary variable $\boldsymbol{\gamma}=\nabla w$ and enforcing $\boldsymbol{\gamma}=\nabla w$ through a Lagrange multiplier in $(\mathrm{Im}\boldsymbol{\Xi})^*=\boldsymbol{H}_{\Gamma}(\mathrm{curl};\Omega)^*$, the authors recover $H^2$-conforming discretizations using standard $H^1$-spaces and curl-conforming spaces. The discrete scheme employs choices of $\tilde{\mathbb{W}}_{\Gamma}$, $\boldsymbol{\mathbb{G}}_{\Gamma}$, and $\boldsymbol{\mathbb{Q}}_{\Gamma}$, together with an iterated penalty method to avoid constructing a basis for $\mathrm{Im}\boldsymbol{\Xi}_X$, and it yields $H^2_{\Gamma}(\Omega)$ solutions that coincide with the original $H^2$-conforming approximations. The approach is demonstrated through 2D and 3D examples, including Kirchhoff plates (static and dynamic) and $H^2$-projections, and extended to time-dependent problems with careful treatment of stability constants. The work also establishes uniform stability results for the Morgan–Scott space and provides right-inverse constructions for curl and the $\Xi_X$ operator, linking to previous work and generalizing it to broader settings.
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.