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Numerical integration of the stiffness matrix in higher-order finite element (FE) methods is recognized as one of the heaviest computational tasks in an FE solver. The problem becomes even more relevant when computing the Gram matrix in the algorithm of the Discontinuous Petrov Galerkin (DPG) FE methodology. Making use of 3D tensor-product shape functions, and the concept of sum factorization, known from standard high-order FE and spectral methods, here we take advantage of this idea for the entire exact sequence of FE spaces defined on the hexahedron. The key piece to the presented algorithms is the exact sequence for the one-dimensional element, and use of hierarchical shape functions. Consistent with existing results, the presented algorithms for the integration of H1, H(curl), H(div), and L2 inner products, have the O(p7) computational complexity in contrast to the O(p9) cost of conventional integration routines. Use of Legendre polynomials for shape functions is critical in this implementation. Three boundary value problems under different variational formulations, requiring combinations of H1, H(div) and H(curl) test shape functions, were chosen to experimentally assess the computation time for constructing DPG element matrices, showing good correspondence with the expected rates. 

We propose and investigate the application of alternative enriched test spaces in the discontinuous Petrov–Galerkin (DPG) finite element framework for singular perturbation linear problems, with an emphasis on 2D convection-dominated diffusion. Providing robust L2 error estimates for the field variables is considered a convenient feature for this class of problems, since this normwould not account for the large gradients present in boundary layers. With this requirement in mind, Demkowicz and others have previously formulated special test norms, which through DPG deliver the desired L2 convergence. However, robustness has only been verified through numerical experiments for tailored test normswhich are problem-specific,whereas the quasi-optimal test norm (not problem specific) has failed such tests due to the difficulty to resolve the optimal test functions sought in the DPG technology. To address this issue (i.e. improve optimal test functions resolution for the quasi-optimal test norm), we propose to discretize the local test spaces with functions that depend on the perturbation parameter ϵ. Explicitly,wework with B-spline spaces defined on an ϵ-dependent Shishkin submesh. Two examples are run using adaptive h-refinement to compare the performance of proposed test spaces with that of standard test spaces. We also include a modified norm and a continuation strategy aiming to improve time performance and briefly experiment with these ideas.