Discontinuous Petrov-Galerkin (DPG) Method with Optimal Test Functions

Leszek Demkowicz <Ten adres pocztowy jest chroniony przed spamowaniem. Aby go zobaczyć, konieczne jest włączenie w przeglądarce obsługi JavaScript.>
Institute for Computational Engineering and Sciences (ICES)
University of Texas at Austin, USA
Progress Report

The success of the DPG method relies on two essential components: on fly computation of optimal test functions, and use of broken test spaces that makes this computation feasible. The method can also be interpreted as a minimum residual approach with the residual measured in a dual norm, or as a mixed problem where one solves simultaneously for (an accurate) representation of the residual [1]. A typical boundary-value problem admits many variational formulations and the DPG methodology can be applied to each of them [1,2]. The method has no preasymptotic behavior [5] and is ''adaptivity ready'' as it comes with a built-in a-posteriori residual error estimator.

On the negative side, similarly to standard DG methods, DPG doubles the number of interface variables, and requires more (than standard Galerkin) expensive element computations.

I will report on the DPG related research in my group in the last two years. The work includes:

- applications of DPG methodology to space-time formulations [3,4],
- construction of preconditioners based on integration with adaptivity [5],
- applications to complex 3D problems [6],
- a solution methodology reducing a DPG problem to a discrete least squares problem, enabling reduction of condition number from 1/h2 to 1/h [7],
- an ultraweak DPG method for polygonal elements of arbitrary shape [8],
- goal-oriented adaptivity for the ultraweak DPG method [9].

[1] L. Demkowicz and J. Gopalakrishnan, Discontinuous Petrov-Galerkin (DPG) Method, Encyclopedia of Computational Mechanics, Second Edition, in print, see also ICES Report 2015/20.

[2] C. Carstensen, L. Demkowicz and J. Gopalakrishnan, Breaking Spaces and Forms for the DPG Method and Applications Including Maxwell Equations, Comput. Math. Appl. 72(3): 494-522, 2016.

[3] T. Ellis, J. Chan and L. Demkowicz, Robust DPG Methods for Transient Convection-Diffusion, in Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Eds. G.R. Barrenechea et al., Lecture Notes in Computational Science and Engineering 114, Spinger, 2016.

[4] L. Demkowicz, J. Gopalakrishnan, S. Nagaraj and P. Sepulveda, A Spacetime DPG Method for the Schrodinger Equation, SIAM J. of Num Anal. accepted, see also ICES Report, 2016/9.

[5] S. Petrides and L. Demkowicz, An Adaptive DPG Method for High Frequency Time-harmonic Wave Propagation Problems, in review, ICES Report 2016/20.

[6] F. Fuentes, L. Demkowicz and A. Wilder, Using a DPG Method to Validate DMA Experimental Calibration of Viscoelastic Materials, in review, see also ICES Report 2017/6.

[7] B. Keith, S. Petrides, F. Fuentes and L. Demkowicz, Discrete Least-squares Finite Element Methods, ICES Report 2017.

[8] A. Vaziri, J. Mora Paz, F. Fuentes and L. Demkowicz, High-Order Polygonal Finite Elements Using Discontinuous Petrov-Galerkin (DPG) Method, ICES Report 2017.

[9] B. Keith, A. Vaziri and L. Demkowicz, Goal-oriented Error Estimation and Adaptive Mesh Refinement in Ultraweak Formulations with the DPG Methodology, ICES Report.