Thesis Summary:

A Hybridizable Discontinuous Galerkin (HDG) method for solving the biharmonic problem $Delta^{2} u=f$ is proposed and analyzed in this work. More precisely, we employ a Discontinuous Galerkin (DG) method based on a system of first-order equations, which we propose to approximate $u$, $ abla u$, $mathcal{H}(u)$ and $ abla cdot mathcal{H}(u)$ simultaneously, where $mathcal{H}$ corresponds to the Hessian matrix. This method allows us to eliminate all the interior variables locally to obtain a global system for $hat{u}_{h}$ and $hat{boldsymbol{q}}_{h}$ that approximate $u$ and $ abla u$, respectively, on the interfaces of the triangulation. As a consequence the only globally coupled degrees of freedom are those of the approximations of $u$ and $ abla u$ on the faces of the elements. We also carry out an priori error analysis using the orthogonal $L^{2}$-projection and conclude that the orders of convergence for the errors in the approximation of $mathcal{H}(u)$, $ abla cdot mathcal{H}(u)$, $ abla u$ and $u$ are $k+1/2$, $k-1/2$, $k$, and $k+1$, respectively, where $k geq 1$ is the polynomial degree of the discrete spaces. Our numerical results suggest that the approximations $mathcal{H}(u)$, $ abla u$ and $u$ converge with optimal order $k+1$ and the approximation of $ abla cdot mathcal{H}(u)$ converge with suboptimal order $k$.