Error representation of the time-marching DPG scheme.

In this article, we introduce an error representation function to perform adaptivity in time of the recently developed time-marching Discontinuous Petrov–Galerkin (DPG) scheme. We first provide an analytical expression for the error that is the Riesz representation of the residual. Then, we approximate the error by enriching the test space in such a way that it contains the optimal test functions. The local error contributions can be efficiently computed by adding a few equations to the time-marching scheme. We analyze the quality of such approximation by constructing a Fortin operator and providing an a posteriori error estimate. The time-marching scheme proposed in this article provides an optimal solution along with a set of efficient and reliable local error contributions to perform adaptivity. We validate our method for both parabolic and hyperbolic problems. • We derive an error representation formula for adaptivity of the time-marching DPG scheme. • We provide the analytical error and an approximation of it by enriching the test space. • We introduce efficient and reliable local error contributions to perform adaptivity. • We analyze the approximation of the error by constructing a global Fortin operator. • We provide an a posteriori error estimate. • We validate our adaptive method for both parabolic and hyperbolic problems. [ABSTRACT FROM AUTHOR]