Convergence rates of the DPG method with reduced test space degree
Provides theoretical justification for observed numerical behavior in DPG methods, benefiting computational scientists using these methods for PDEs.
The paper proves a duality theorem for DPG methods explaining higher convergence rates in weaker norms, and shows that the DPG method remains solvable with reduced odd test space degree for the Laplace equation, with a non-conforming analysis matching numerical rates.
This paper presents a duality theorem of the Aubin-Nitsche type for discontinuous Petrov Galerkin (DPG) methods. This explains the numerically observed higher convergence rates in weaker norms. Considering the specific example of the mild-weak (or primal) DPG method for the Laplace equation, two further results are obtained. First, the DPG method continues to be solvable even when the test space degree is reduced, provided it is odd. Second, a non-conforming method of analysis is developed to explain the numerically observed convergence rates for a test space of reduced degree.