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Advanced Numerical Methods in the Mathematical Sciences

Leszek Demkowicz, University of Texas at Austin
Discontinuous Petrov-Galerkin (DPG) Method with Optimal Test Functions: Tutorial and Perspectives

Authors

  • Leszek Demkowicz, University of Texas at Austin
  • Jay Gopalakrishnan, Portland State University

Abstract

The DPG method wears three hats [2]. It is a Petrov-Galerkin method with optimal test functions computed on the fly to reproduce stability properties of the continuous variational formulation [1]. It is also a minimum-residual method with the residual measured in a dual norm corresponding to a user-defined test norm [1,2]. This gives (an indirect) possibility of controlling the norm of convergence, a feature especially attractive in context of constructing robust discretizations for singular perturbation problems [4,5,9]. Finally, it is also a mixed method in which one simultaneously solves for the residual [3,2], which provides an excellent a-posteriori error estimator and enables automatic adaptivity.

In practice, the optimal test functions and residual are approximated within a finite-dimensional enriched test space. The mixed method framework provides a natural starting point for analyzing effects of such an approximation through the construction of appropriate Fortin operators [7,4].

What makes the whole story possible is the use of discontinuous test functions (broken test spaces). The trial functions may, but need not be, discontinuous. The paradigm of "breaking" test functions can be applied to any well-posed variational formulation [8,4] including well known classical and mixed formulations and the less known ultraweak formulation. When applied to trivial (strong) variational formulation, DPG reduces to the well known least-squares method. DPG results in a hybridization of the original formulation where one solves additionally for fluxes (and traces in the ultraweak case) on the mesh skeleton. The hybridization approximately doubles the number of interface unknowns when compared with classical conforming elements or HDG methods but it is comparable with a number of unknowns for other DG formulations. Computation of optimal test functions and residual is done locally, at the element level. Thus, it does not contribute to the cost of the global solve (but it is significant for systems of 3D equations).

The DPG methodology guarantees a stable discretization for any well-posed boundary- or initial boundary-value problem (space-time formulations). In particular, it can be applied to all problems where the standard Galerkin fails or is stable only in the asymptotic regime. Being a Ritz method, DPG enables adaptive computations starting with coarse meshes. For instance, for wave propagation problems, the initial mesh need not even satisfy the Nyquist criterion.

The presentation will consist of two parts. First, the tutorial will provide an overview of the main points made above, illustrated with 1D and 2D numerical examples. The second part will outline some open problems and some of our current work on the subject.

References

  1. 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.
  2. L. Demkowicz and J. Gopalakrishnan, An Overview of the DPG Method, in Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Different Equations, eds: X. Feng, O. Karakashian, Y. Xing, IMA Publications, Springer-Verlag ().
  3. A. Cohen, W. Dahmen, and G. Welper, Adaptivity and Variational Stabilization for Convection-Diffusion Equations, ESAIM Math. Model. Numer. Anal. (), 46(5): 1247-1273.
  4. L. Demkowicz, C. Carstensen, and J. Gopalakrishnan, The Paradigm of Broken Test Functions in DPG Discretizations, in preparation.
  5. L. Demkowicz and H. Heuer, Robust DPG Method for Convection-Dominated Diffusion Problems, SIAM J. Num. Anal. (), 51: 2514-2537.
  6. 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. (), 67(4): 771-795.
  7. J. Gopalakrishnan and W. Qiu, An Analysis of the Practical DPG Method, Math. Comp. () 83(286): 537-552.
  8. L. Demkowicz, Various Variational Formulations and Closed Range Theorem, ICES Report 15-03.
  9. L. Demkowicz and I. Harari, Robust Discontinuous Petrov Galerkin (DPG) Methods for Reaction-Dominated Diffusion, ICES Report 14-36.