Alternative Enriched Test Spaces in the DPG Method for Singular Perturbation Problems
Alternative Enriched Test Spaces in the DPG Method for Singular Perturbation Problems
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DOI：
10.1515/cmam-2018-0207
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发表时间：
2019-04
影响因子：
1.3
通讯作者：
J. Salazar;Jaime Mora;L. Demkowicz
中科院分区：
文献类型：
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作者：
J. Salazar;Jaime Mora;L. Demkowicz
摘要本文提出并研究了在不连续Petrov-Galerkin（DPG）有限元框架中交替丰富测试空间在奇异摄动线性问题中的应用，重点是二维对流占优扩散。为场变量提供鲁棒的L2{L^{2}}误差估计被认为是这类问题的一个方便的特征，因为该范数不会考虑边界层中存在的大梯度。考虑到这一要求，Demkowicz和其他人以前制定了特殊的测试规范，通过DPG提供所需的L2{L^{2}}收敛。然而，鲁棒性仅通过针对特定问题的定制测试规范的数值实验进行了验证，而准最佳测试规范（不是特定问题）由于难以解决DPG技术中寻求的最佳测试函数而未能通过此类测试。为了解决这个问题（即提高准最优测试范数的最优测试函数分辨率），我们建议用依赖于扰动参数的函数来离散局部测试空间。明确地说，我们的工作与B样条空间上定义的一个依赖于Shishkin子网格。两个例子运行使用自适应h-细化建议的测试空间与标准测试空间的性能进行比较。我们还包括一个修改后的规范和连续的策略，旨在提高时间性能，并简要地实验这些想法。
Abstract 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{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 L2{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.