A low-order discontinuous Petrov–Galerkin method for the Stokes equations

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Abstract

This paper introduces a low-order discontinuous Petrov-Galerkin (dPG) finite element method (FEM) for the Stokes equations. The ultra-weak formulation utilizes piecewise constant and affine ansatz functions and piecewise affine and discontinuous lowest-order Raviart–Thomas test search functions. This low-order discretization for the Stokes equations allows for a direct proof of the discrete inf-sup condition with explicit constants. The general framework of Carstensen et al. (SIAM J Numer Anal 52(3):1335–1353, 2014) then implies a complete a priori and a posteriori error analysis of the dPG FEM in the natural norms. Numerical experiments investigate the performance of the method and underline its quasi-optimal convergence.

Keywords

Stokes Discontinuous Petrov Galerkin Low-order discretization A priori A posteriori Adaptive mesh refinement 

Mathematics Subject Classification

65N12 65N15 65N30 65Y05 65Y20 

Notes

Acknowledgements

This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) in the Priority Program 1748 ‘Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis’ under the project CA 151/22-1. The second author is supported by the Berlin Mathematical School.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHumboldt-Universität zu BerlinBerlinGermany

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