We propose a three dimensional Discontinuous Petrov-Galerkin Maxwell approach for modeling Raman gain in fiber laser amplifiers. In contrast with popular beam propagation models, we are interested in a truly full vectorial approach. We apply the ultraweak DPG formulation, which is known to carry desirable properties for high-frequency wave propagation problems, to the coupled Maxwell signal/pump system and use a nonlinear iterative scheme to account for the Raman gain. This paper also introduces a novel and practical full-vectorial formulation of the electric polarization term for Raman gain that emphasizes the fact that the computer modeler is only given a measured bulk Raman gain coefficient. Our results provide promising qualitative corroboration of the model and methodology used.
Numerical integration of the stiffness matrix in higher order finite element (FE) methods is recognized as one of the heaviest computational tasks in a 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 H 1 , H(curl), H(div), and L 2 inner products, have the O(p 7 ) computational complexity. Additionally, a modified version of the algorithms is proposed when the element map can be simplified, resulting in the reduced O(p 6 ) complexity. Use of Legendre polynomials for shape functions is critical in this implementation. Computational experiments performed with H 1 , H(div) and H(curl) test shape functions show 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 {L^{2}} error estimates for the field variables is considered a convenient feature for this class of problems, since this norm would 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 {L^{2}} convergence. However, robustness has only been verified through numerical experiments for tailored test norms which 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, we work 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.
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