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Discontinuous Petrov Galerkin (DPG) Method with Optimal Test Functions

Leszek Demkowicz

Discontinuous Petrov Galerkin (DPG) Method with Optimal Test Functions

pdfpart1.pdf2.32 MB
pdfpart2.pdf22.69 MB


The coming June will mark the sixth anniversary of the first two papers in which Jay Gopalakrishnan and I proposed a novel Finite Element (FE) technology based on what we called the ``ultra-weak variational formulation'' and the idea of computing (approximately) optimal test functions on the fly [1,2,3]. We called it the ``Discontinuous Petrov Galerkin Method''. Shortly afterward we learned that we owned neither the concept of the ultra-weak formulation nor the name of the DPG method, both introduced in a series of papers by colleagues from Milano: C. L. Bottasso, S. Micheletti, P. Causin and R. Sacco, several years earlier. The name ``ultra-weak'' was stolen from O. Cessenat and B. Despres. But the idea of computing optimal test functions was new... From the very beginning we were aware of the fact that the Petrov-Galerkin formulation is equivalent to a Minimum Residual Method (generalized Least Squares) in which the (minimized) residual is measured in a dual norm, the idea pursued much earlier by colleagues from Texas A&M: J. Bramble, R. Lazarov and J. Pasciak. Jay and I were lucky; a few months after putting [1,2] on line, Wolfgang Dahmen and Chris Schwab presented essentially the same approach pointing to a connection with mixed methods and the fact that the use of discontinuous test functions is not necessary.

So, six years, over 20 papers and three Ph.D. dissertations later, I will attempt to summarize in the three lectures the fundamentals, present extensive numerical results and outline the current frontier. For more up-to-date information, visit


for presentations given during the first Least Squares/DPG workshop organized at ICES in November 2013.

Lecture 1: The DPG method guarantees stability for any well-posed linear


We will discuss the equivalence of several formulations: Petrov-Galerkin method with optimal test functions, minimum residual formulation and a mixed formulation. We will summarize well-posedness results for formulations with broken test functions: the ultra-weak formulation based on first order systems and the formulation derived from standard second order equations. Standard model problems: Poisson, linear elasticity, Stokes, linear acoustics and Maxwell equations, will be used to illustrate the methodology with h-, p-, and hp-convergence tests. The DPG method comes with a posteriori-error evaluator (not estimator...) built in which provides a natural framework for adaptivity.

Lecture 2: Singular perturbation problems. Extrapolation to nonlinear


The DPG methodology allows for controlling the norm in which we want to converge by selecting the right norm for residual. I will show how the idea translates into superb stability properties (and eliminates the need for stabilization) for convection dominated problems: convection-dominated diffusion, incompressible and compressible Navier Stokes equations [4,5,6,7].

[1] L. Demkowicz and J. Gopalakrishnan, ``A class of discontinuous Petrov-Galerkin methods. PartI: The transport equation,'' CMAME: 199, 23-24, 1558-1572, 2010.

[2] L. Demkowicz and J. Gopalakrishnan,

``A class of discontinuous Petrov-Galerkin methods. Part II: Optimal test functions,'' Num. Meth. Part. D.E.:27, 70-105, 2011.

[3] ``An Overview of the DPG Method , ICES Report 2013/2, also in ``Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations'', eds: X. Feng, O. Karakashian, Y. Xing, IMA Publications, Springer-Verlag, 2013.

[4] L. Demkowicz and N. Heuer,``Robust DPG Method for Convection-Dominated Diffusion Problems'', SIAM J. Num. Anal 51: 2514-2537, 2013, see also ICES Report 2011/13.

[5] J. Chan, N, Heuer, T Bui-Thanh and L. Demkowicz, `` Robust DPG Method for Convection-dominated Diffusion Problems II: Natural Inflow Condition'', Comput. Math. Appl.,2013, in print, see also ICES Report 2012/21.

[6] Nathan Roberts. ``A Discontinuous Petrov-Galerkin Methodology for Incompressible Flow Problems'', PhD thesis, University of Texas at Austin, August 2013. (supervisors: L. Demkowicz and R. Moser).

[7] Jesse Chan,``A DPG Method for Convection-Diffusion Problems'', PhD thesis, University of Texas at Austin, July 2013 (supervisors: L. Demkowicz and R. Moser).

© Realizacja Zbigniew Kacprzyk