# A posteriori error estimation of $\mathit{hp}$-dG finite element methods for highly indefinite Helmholtz problems

Sauter, Stefan A; Zech, J (2015). A posteriori error estimation of $\mathit{hp}$-dG finite element methods for highly indefinite Helmholtz problems. SIAM Journal on Numerical Analysis, 53(5):2414-2440.

## Abstract

In this paper, we will consider an $\mathit{hp}$-finite elements discretization of a highly indefinite Helmholtz problem by some dG-formulation which is based on the ultra-weak variational formulation by Cessenat and Deprés. We will introduce an a posteriori error estimator and derive reliability and efficiency estimates which are explicit with respect to the wavenumber and the discretization parameters $\mathit{h}$ and $\mathit{p}$. In contrast to the conventional conforming finite element method for indefinite problems, the dG-formulation is unconditionally stable and the adaptive discretization process may start from a very coarse initial mesh. Numerical experiments will illustrate the efficiency and robustness of the method.

## Abstract

In this paper, we will consider an $\mathit{hp}$-finite elements discretization of a highly indefinite Helmholtz problem by some dG-formulation which is based on the ultra-weak variational formulation by Cessenat and Deprés. We will introduce an a posteriori error estimator and derive reliability and efficiency estimates which are explicit with respect to the wavenumber and the discretization parameters $\mathit{h}$ and $\mathit{p}$. In contrast to the conventional conforming finite element method for indefinite problems, the dG-formulation is unconditionally stable and the adaptive discretization process may start from a very coarse initial mesh. Numerical experiments will illustrate the efficiency and robustness of the method.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2015 10 Aug 2016 08:19 10 Aug 2016 08:19 Society for Industrial and Applied Mathematics 0036-1429 https://doi.org/10.1137/140973955 http://dx.doi.org/10.1137/140973955 (Publisher)