Discontinuous Petrov-Galerkin (DPG) Method with Optimal Test Functions
Despite some more than 10 years of the development, the DPG method still surprises us as we keep learning new, fundamental facts about it. The three two hour lectures comprising the tutorial aim at covering fundamentals of the method and flashing some representative numerical examples contrasting the method with standard Galerkin schemes. Both theory and numerics will be illustrated with classical model problems: diffusion-convection-reaction and time-harmonic elastodynamics, acoustics and Maxwell equations.
Lecture 1: Tutorial on classical Galerkin Finite Element (FE) methods
It is impossible to understand the DPG method w/o a background in the classical FE analysis. The first lecture will cover classical notions and facts from Numerical Analysis:
a) Variational formulations leading to different energy spaces;
b) The exact sequence energy spaces: H^1,H(curl),H(div)and L^2 spaces; the corresponding conformity requirements;
c) Babuska and Brezzi theories and the relation between them.
Lecture 2: Variational formulations with broken test spaces
Each of the variational formulations discussed in the first lecture may serve as a starting point for a version with broken (mesh dependent) test functions. Use of broken test spaces is critical for the application of the DPG technology and comes at the price of introducing additional global unknowns: traces and/or fluxes at the mesh skeleton. Surprisingly perhaps, all formulations
share the same number of interface unknowns. We shall discuss the well-posedness of such formulations.
Lecture 3: Three interpretations of the DPG method
The DPG method wears ''three hats'' and can be interpreted as a Petrov-Galerkin scheme with (optimal) test functions computed on the fly, a Minimum Residual Method, and a Mixed Method where one solves simultaneously for the approximate solution and Riesz representation of the residual. Critical to all three interpretations is the inversion of Riesz operator on local, element level. In practice, the Riesz operator has to be approximated, either a-priori (concept of ``enriched test space'') or a-posteriori, in an adaptive way. The DPG method comes with an a-posteriori error estimation built in and provides a setting for adaptivity from day one.