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This article summarizes the Lp-DPG method presented in [18], where only 1D convection-diffusion problems are solved.We apply the same computational techniques to 2D convection-diffusion problems and report additional numerical results herein. Furthermore, we propose an Lp-DPG method with variable p and illustrate it with numerical experiments. 

In this article, we introduce an error representation function to perform adaptivity in time of the recently developed timemarching 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. 

The Discontinuous Petrov-Galerkin (DPG) method and the exponential integrators are two well establishednumerical methods for solving Partial Differential Equations (PDEs) and stiff systems of Ordinary Differential Equations (ODEs), respectively. In this work, we apply the DPG method in the time variable for linear parabolic problems and we calculate the optimal test functions analytically. We show that the DPG method in time is equivalent to exponential integrators for the trace variables, which are decoupled from the interior variables. In addition, the DPG optimal test functions allow us to compute the approximated solutions in the time element interiors. This DPG method in time allows to construct a posteriori error estimations in order to perform adaptivity. We generalize this novel DPG-based time-marching scheme to general first order linear systems of ODEs. We show the performance of the proposed method for 1D and 2D +time linear parabolic PDEs after discretizing in space by the finite element method. 

Following Muga and van der Zee (Muga and van der Zee, 2015), we generalize the standard Discontinuous Petrov–Galerkin (DPG) method, based on Hilbert spaces, to Banach spaces. Numerical experiments using model 1D convection-dominated diffusionproblem are performed and compared with Hilbert setting. It is shown that Banach basedmethod gives solutions less susceptible to Gibbs phenomenon. h-adaptivity is implemented with the help of the error representation function as error indicator. 

We construct a general family of DPG Fortin operators for the exact energy spaces defined on a tetrahedral element. 

Higher order finite element (FE) methods provide significant advantages in a number of applications such as wave propagation, where high order shape functions help to mitigate pollution (dispersion) error. However, classical assembly of higher order systems is computationally burdensome, requiring the evaluation of many point quadrature schemes. When the Discontinuous Petrov-Galerkin (DPG) FE methodology is employed, the use of an enriched test space further increases the computational burden of system assembly, increasing the relevance of improved assembly techniques. Sum factorization—a technique that exploits the tensorproduct structure of shape functions to accelerate numerical integration—was proposed in Ref. [10] for the assembly of DPG matrices on hexahedral elements that reduced the computational complexity from order (p9) to (p7) (where p denotes polynomial order). In this work we extend the concept of sum factorization to the construction of DPG matrices on prismatic elements by expressing prism shape functions as tensor products of 2D simplex and 1D interval shape functions. Unexpectedly, the resulting sum factorization routines on partially-tensorized prism shape functions achieve the same (p7) complexity as sum factorization on fully-tensorized hexahedra shape functions (as products of 1D interval shape functions) presented in Ref. [10]. Throughout this work we adhere to the theory of exact sequence energy spaces, proposing sum factorization routines for each of the 3D FE exact sequence energy spaces—H1, H(curl), H(div), and L2. Computational results for construction of the DPG Gram matrix on a prismatic element in each exact sequence energy space are presented, corroborating the expected (p7) complexity. Additionally, construction of the DPG system for an ultraweak Maxwell problem on a prismatic element is considered and a partially-tensorized sum factorization for hexahedral elements is proposed to improve implementational compatibility between hexahedral and prismatic elements. 

This article introduces the DPG-star finite element method.