hpfem/esco2012-boa

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 \title{$hp$-Adaptive Eigensolution: Benchmark Studies} \tocauthor{H. Hakula} \author{} \institute{} \maketitle \begin{center} {\large \underline{Harri Hakula}}\\ Aalto University\\ {\tt Harri.Hakula@aalto.fi} \\ \vspace{4mm}{\large Tomi Tuominen}\\ Aalto University\\ {\tt tatuomin@math.hut.fi} \end{center} \section*{Abstract} In this paper we discuss the use of $hp$-adaptive methods of Houston \& S\"uli-type \cite{HSS} in the context of eigenproblems. Using a combination of bubble modes and Sobolev-regularity estimation we arrive at an adaptive algorithm controlling both $h$ and $p$. In broad terms our work can also be interpreted as an extension of \cite{GO} to high-order FEM. For an eigenpair $(\mu,\textbf{u})$ we define a discrete variational problem (omitting the customary use of $h$): Find $\textbf{u} \in V$ and $\mu \in \mathbb{R}$ such that $a(\textbf{u},\textbf{v}) = \mu\,m(\textbf{u},\textbf{v}), \forall\, \textbf{v} \in V$, where $a(\cdot,\cdot)$ and $m(\cdot,\cdot)$ are the bilinear forms of stiffness and mass matrices, respectively. Let us assume that $V$ corresponds to standard elements of $p=4$. Then we consider an extension $V^+$ with bubbles for $p=5,6,7$, say, and compute a local (elemental) error estimator $\boldsymbol{\epsilon}$: Find $\boldsymbol{\epsilon} \in V^+$ such that $a(\textbf{u}+\boldsymbol{\epsilon},\textbf{v}) = \mu\,m(\textbf{u},\textbf{v}), \forall\, \textbf{v} \in V^+$, i.e., $a(\boldsymbol{\epsilon},\textbf{v})~=~\mu\,m(\textbf{u},\textbf{v}) - a(\textbf{u},\textbf{v})$. In this work we present results on two benchmark problems: isospectral drums (Laplace operator) and rotor blade (piece of a cylindrical shell). These results are then compared with the best a priori results in both cases. \bibliographystyle{plain} \begin{thebibliography}{10} \bibitem{HSS} {\sc P. {Houston} and B. {Senior} and E. {S\"uli}}. {Sobolev regularity estimation for $hp$-adaptive finite element methods}. Proceedings of ENUMATH 2001, pp. 631--656.. \bibitem{GO} {\sc L. Grubisic and J. S. Ovall.}. {On estimators for eigenvalue/eigenvector approximations}. {Math. Comp.}, 78 (2009), 739--770. \end{thebibliography}