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Volume 27, Issue 6
A New Approach to Recovery of Discontinuous Galerkin

Sebastian Franz, Lutz Tobiska & Helena Zarin

DOI: 10.4208/jcm.2009.09-m2899

J. Comp. Math., 27 (2009), pp. 697-712.

Published online: 2009-12

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  • Abstract

A new recovery operator $P : Q^{disc}_n (\mathcal{T}) → Q^{disc}_{n+1}(\mathcal{M})$ for discontinuous Galerkin is derived. It is based on the idea of projecting a discontinuous, piecewise polynomial solution on a given mesh $\mathcal{T}$ into a higher order polynomial space on a macro mesh $\mathcal{M}$. In order to do so, we define local degrees of freedom using polynomial moments and provide global degrees of freedom on the macro mesh. We prove consistency with respect to the local $L_2$-projection, stability results in several norms and optimal anisotropic error estimates. As an example, we apply this new recovery technique to a stabilized solution of a singularly perturbed convection-diffusion problem using bilinear elements.

  • Keywords

Discontinuous Galerkin, Postprocessing, Recovery.

  • AMS Subject Headings

65N12, 65N15, 65N30.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-27-697, author = {Sebastian Franz, Lutz Tobiska and Helena Zarin}, title = {A New Approach to Recovery of Discontinuous Galerkin}, journal = {Journal of Computational Mathematics}, year = {2009}, volume = {27}, number = {6}, pages = {697--712}, abstract = {

A new recovery operator $P : Q^{disc}_n (\mathcal{T}) → Q^{disc}_{n+1}(\mathcal{M})$ for discontinuous Galerkin is derived. It is based on the idea of projecting a discontinuous, piecewise polynomial solution on a given mesh $\mathcal{T}$ into a higher order polynomial space on a macro mesh $\mathcal{M}$. In order to do so, we define local degrees of freedom using polynomial moments and provide global degrees of freedom on the macro mesh. We prove consistency with respect to the local $L_2$-projection, stability results in several norms and optimal anisotropic error estimates. As an example, we apply this new recovery technique to a stabilized solution of a singularly perturbed convection-diffusion problem using bilinear elements.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2009.09-m2899}, url = {http://global-sci.org/intro/article_detail/jcm/8598.html} }
TY - JOUR T1 - A New Approach to Recovery of Discontinuous Galerkin AU - Sebastian Franz, Lutz Tobiska & Helena Zarin JO - Journal of Computational Mathematics VL - 6 SP - 697 EP - 712 PY - 2009 DA - 2009/12 SN - 27 DO - http://doi.org/10.4208/jcm.2009.09-m2899 UR - https://global-sci.org/intro/article_detail/jcm/8598.html KW - Discontinuous Galerkin, Postprocessing, Recovery. AB -

A new recovery operator $P : Q^{disc}_n (\mathcal{T}) → Q^{disc}_{n+1}(\mathcal{M})$ for discontinuous Galerkin is derived. It is based on the idea of projecting a discontinuous, piecewise polynomial solution on a given mesh $\mathcal{T}$ into a higher order polynomial space on a macro mesh $\mathcal{M}$. In order to do so, we define local degrees of freedom using polynomial moments and provide global degrees of freedom on the macro mesh. We prove consistency with respect to the local $L_2$-projection, stability results in several norms and optimal anisotropic error estimates. As an example, we apply this new recovery technique to a stabilized solution of a singularly perturbed convection-diffusion problem using bilinear elements.

Sebastian Franz, Lutz Tobiska and Helena Zarin. (2009). A New Approach to Recovery of Discontinuous Galerkin. Journal of Computational Mathematics. 27 (6). 697-712. doi:10.4208/jcm.2009.09-m2899
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