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The Discontinuous Petrov Galerkin Method

Please note that this page is old.
Check in the VVZ for a current information.

Lecturer/Supervisor: Prof. Ralf Hiptmair

Assistant: Cecilia Pagliantini

Time and venue: Friday, 08-10, HG F 26.3
Prep meeting: Friday, Feb 26 at 08:15 in HG F 26.3

Catalogue data: see the VVZ webpage

The number of participants of the seminar is limited to 10.

Schedule:

     March 4, 2016
     April 8, 2016
     April 29, 2016
     May 6, 2016
     May 13, 2016
     May 20, 2016
     May 27, 2016


Abstract

The new Discontinuous Petrov Galerkin Method (DPG) is a generalized finite element approach pursuing the idea of choosing approximately optimal test functions in (piecewise polynomial) spaces with relaxed continuity requirements.

The benefit is enhanced stability of the discrete variational formulations, which is particularly important for singularly perturbed problems. In fact, the discontinuous Petrov Galerkin (DPG) method can be interpreted as three apparently different, but equivalent methods: a method with a test space computed on the fly, a method that minimizes a residual in a dual norm, and a mixed method with nonstandard but stable pair of spaces.

Aim and content of the seminar

Studying DPG the students should learn about general concepts and numerical analysis techniques relevant for the discretization of boundary value problems for linear partial differential equations.

The seminar will comprise presentations based on key scientific publications about the DPG method.

The seminar will comprise up to 10 student presentations of a duration of about 60 minutes. They should be partly based on PDF slides prepared using the BEAMER LATEXpackage (or LATEXbased tools under MacOS). The presentations should be done using a laptop computer (which can be provided). Speakers are advised to elaborate technical manipulations and proofs on the blackboard. The lecture slides in PDF format should be made available immediately after the presentation.

Quiz

Participants of the seminar will be asked questions about the previous presentations at the beginning of each session.

Introduction to the topic

Survey paper:

[DG14] L. Demkowicz and J. Gopalakrishnan. An overview of the discontinuous Petrov Galerkin method. In Xiaobing Feng, Ohannes Karakashian, and Yulong Xing, editors, Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, volume 157 of The IMA Volumes in Mathematics and its Applications, pages 149–180. Springer International Publishing, 2014.

ICERM lecture by Prof. J. Gopalakrishnan: A priori and a posteriori analyses of DPG methods

References

[BGH14] T. Bouma, J. Gopalakrishnan, and A. Harb. Convergence rates of the DPG method with reduced test space degree. Comput. Math. Appl., 68(11):1550–1561, 2014.

[BTDG13] T. Bui-Thanh, L. Demkowicz, and O. Ghattas. A unified discontinuous Petrov-Galerkin method and its analysis for Friedrichs’ systems. SIAM J. Numer. Anal.,  51(4):1933–1958, 2013.

[CDG14] C. Carstensen, L. Demkowicz, and J. Gopalakrishnan. A posteriori error control for DPG methods. SIAM J. Numer. Anal., 52(3):1335–1353, 2014.

[CDG15] C. Carstensen, L. Demkowicz, and J. Gopalakrishnan. Breaking spaces and forms for the DPG method and applications including Maxwell equations. Numer. Math., 2015.

[CDW12] A. Cohen, W. Dahmen, and G. Welper. Adaptivity and variational stabilization for convection-diffusion equations. ESAIM Math. Model. Numer. Anal., 46(5):1247–1273, 2012.

[CEQ14] J. Chan, J. A. Evans, and W. Qiu. A dual Petrov-Galerkin finite element method for the convection-diffusion equation. Comput. Math. Appl., 68(11):1513–1529, 2014.

[DG11] L. Demkowicz and J. Gopalakrishnan. Analysis of the DPG method for the Poisson equation. SIAM J. Numer. Anal., 49(5):1788–1809, 2011.

[DG13] L. Demkowicz and J. Gopalakrishnan. A primal DPG method without a first-order reformulation. Comput. Math. Appl., 66(6):1058–1064, 2013.

[DG14] L. Demkowicz and J. Gopalakrishnan. An overview of the discontinuous Petrov Galerkin method. In Xiaobing Feng, Ohannes Karakashian, and Yulong Xing, editors, Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, volume 157 of The IMA Volumes in Mathematics and its Applications, pages 149–180. Springer International Publishing, 2014.

[DH13] L. Demkowicz and N. Heuer. Robust DPG method for convection-dominated diffusion problems. SIAM J. Numer. Anal., 51(5):2514–2537, 2013.

[GQ14] J. Gopalakrishnan and W. Qiu. An analysis of the practical DPG method. Math. Comp., 83(286):537–552, 2014.

[RBTD14] N. V. Roberts, T. Bui-Thanh, and L. Demkowicz. The DPG method for the Stokes problem. Comput. Math. Appl., 67(4):966–995, 2014.

[SS10] S. Sauter and C. Schwab. Boundary Element Methods, Volume 39 of Springer Series in Computational Mathematics. Springer, Heidelberg, 2010.

 

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