Combining the DPG method with finite elements
Thomas Führer, Norbert Heuer, Michael Karkulik, and Rodolfo Rodríguez
Comput. Methods Appl. Math. 18 (4), 639-652, 2018.
We propose and analyze a discretization scheme that combines the discontinuous Petrov-Galerkin
and finite element methods. The underlying model problem is of general diffusion-advection-reaction
type on bounded domains, with decomposition into two sub-domains. We propose
a heterogeneous variational formulation that is of the ultra-weak (Petrov-Galerkin)
form with broken test space in one part, and of Bubnov-Galerkin form in the other.
A standard discretization with conforming approximation spaces and appropriate test spaces
(optimal test functions for the ultra-weak part and standard test functions for the Bubnov-Galerkin
part) gives rise to a coupled DPG-FEM scheme. We prove its well-posedness and quasi-optimal
convergence. Numerical results confirm expected convergence orders.