In this work we give a complete error analysis of the Discontinuous Petrov
Galerkin (DPG) method, accounting for all the approximations made in its
practical implementation. Specifically, we consider the DPG method that uses a
trial space consisting of polynomials of degree p on each mesh element.
Earlier works showed that there is a "trial-to-test" operator T, which when
applied to the trial space, defines a test space that guarantees stability. In
DPG formulations, this operator T is local: it can be applied
element-by-element. However, an infinite dimensional problem on each mesh
element needed to be solved to apply T. In practical computations, T is
approximated using polynomials of some degree r>p on each mesh element. We
show that this approximation maintains optimal convergence rates, provided that
r≥p+N, where N is the space dimension (two or more), for the Laplace
equation. We also prove a similar result for the DPG method for linear
elasticity. Remarks on the conditioning of the stiffness matrix in DPG methods
are also included.Comment: Mathematics of Computation, 201