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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.
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An Lp-DPG Method with Application to 2D Convection-Diffusion Problems
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.
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- Award ID(s):
- 1819101
- PAR ID:
- 10339105
- Date Published:
- Journal Name:
- Computational methods in applied mathematics
- Volume:
- 22
- Issue:
- 3
- ISSN:
- 1609-4840
- Page Range / eLocation ID:
- 649–662
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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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.more » « less
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