An Overview of the Discontinuous Petrov Galerkin Method
We discuss our current understanding of the discontinuous Petrov Galerkin (DPG) Method with Optimal Test Functions and provide a literature review on the subject.
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Introduction. The adventure with the discontinuous Petrov Galerkin (DPG) method started in Spring 2009. Analyzing spectral methods for the simplest 1D convection problem, we realized that the choice of test function v = u leading to the standard DG method was far from an optimal one in terms of implied stability properties [23]. The main breakthrough came with a realization that the use of ultraweak variational formulation and discontinuous test functions allowed for the computation of (approximate) optimal test functions [25]. After reporting the exciting results in a Mafelap plenary talk, in June 2009, we had learned that we owned neither the concept of the ultraweak formulation nor even the name—the DPG method. Both were introduced several years earlier by the Italian colleagues [4, 5, 12, 13].1 But the concept of computing the optimal test functions on the fly was new, and we pursued a numerical implementation of hp-adaptivity quickly in [27] demonstrating the superior stability properties of the new method. We devoted a considerable amount of our time and resources to the DPG research in the next three years. As it usually happens, our understanding did not grow in a systematic “monotone” mode and, hence, attempting to follow the DPG work in a chronological order would rather be confusing. Instead we present a review of the main concepts behind the DPG methodology as we understand them today: minimization of residuals in dual norms in Sect. 2, use of discontinuous test functions in Sect. 3, ultraweak variational formulations in Sect. 4, selection of optimal test norm for singular perturbation problems in Sect. 5, and the important interpretation of the DPG method as a localization of the PG method with ∗ Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USA, [email protected] † Department of Mathematics, Portland State University, Portland, OR 97207, USA, [email protected] 1 To our credit, the ultraweak formulation was used at that point very formally, without a proper Functional Analysis setting which we established later in [24].
X. Feng et al. (eds.), Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, The IMA Volumes in Mathematics and its Applications 157, DOI 10.1007/978-3-319-01818-8 6, © Springer International Publishing Switzerland 2014
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Leszek Demkowicz and Jay Gopalakrishnan
global optimal test functions in Sect. 6. We conclude with an outline of our current work in Sect. 7. The work on the DPG methodology has barely begun and we hope that more colleagues will get interested in the subject and join us in this endeavor. 2. DPG Is a Minimum Residual Method. DPG methods, like least squares methods, belong to the class of minimum residual methods. We start with a (linear) variational problem, u∈U (2.1) b(u, v) = l(v) ∀v ∈ V . Here U is a trial space and V is a test space. We shall assume that both U and V are Hilbert spaces, b is a sesquilinear (bilinear in the real case) and continu
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