# Undergraduate Thesis of Alejandra Barrios

Career | Mathematical Civil Engineering, Universidad de Concepción | |
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Enrollment Year | 2011 | |

Senior Year | 2018 | |

Thesis Title | A discontinuous Galerkin method for the biharmonic problem | |

## 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 $k$. | ||

Thesis Director(s) | Manuel Solano | |

Thesis Project Approval Date | 2017, March 01 | |

Thesis Defense Date | 2018, November 19 | |

Professional Monitoring | ||

PDF Tesis | Download Thesis PDF | |

(No publications) |

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