A posteriori error estimates for discontinuous Galerkin methods using non-polynomial basis functions. Part II: Eigenvalue problems
University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；RWTH Aachen University（亚琛工业大学）；Forschungszentrum Jülich（于利希研究中心）
Lin Lin, Benjamin Stamm
备注：
arXiv admin note: text overlap with arXiv:1502.01738
我们提出了一个系统的工作，用于推导一般非多项式基函数在内罚不连续伽辽金（DG）格式中求解与二阶线性算子相关的特征值问题的后验误差估计。此类特征值问题在科学和工程应用中起重要作用，特别是在理论化学、固态物理和材料科学中。基于在[{\it L. Lin, B. Stamm, http://dx.doi.org/10.1051/m2an/2015069}]中为二阶PDEs开发的框架，我们为特征值问题开发了残差型上界和下界误差估计，用于测量后验误差。我们方法的主要优点是该方法参数无关，即上界和下界估计中除了一个依赖于解的常数外，其余所有常数都可以通过求解局部和独立的特征值问题显式计算，而唯一的不可计算的常数可以合理地用一个可计算的常数近似，而不影响估计的整体有效性。与PDE的情况相比，我们发现特征值问题的后验误差估计器必须忽略某些项，这些项涉及精确的特征值或特征函数，这些在数值模拟中是不可获得的。我们仔细定义了这些项，并通过数值方法证明被忽略的项确实比可计算的估计器高阶。各种一维和二维问题的数值结果表明，上界和下界均能有效测量特征值和特征函数的误差。
英文摘要：
We present the first systematic work for deriving a posteriori error estimates for general non-polynomial basis functions in an interior penalty discontinuous Galerkin (DG) formulation for solving eigenvalue problems associated with second order linear operators. Eigenvalue problems of such types play important roles in scientific and engineering applications, particularly in theoretical chemistry, solid state physics and material science. Based on the framework developed in [{\it L. Lin, B. Stamm, http://dx.doi.org/10.1051/m2an/2015069}] for second order PDEs, we develop residual type upper and lower bound error estimates for measuring the a posteriori error for eigenvalue problems. The main merit of our method is that the method is parameter-free, in the sense that all but one solution-dependent constants appearing in the upper and lower bound estimates are explicitly computable by solving local and independent eigenvalue problems, and the only non-computable constant can be reasonably approximated by a computable one without affecting the overall effectiveness of the estimates in practice. Compared to the PDE case, we find that a posteriori error estimators for eigenvalue problems must neglect certain terms, which involves explicitly the exact eigenvalues or eigenfunctions that are not accessible in numerical simulations. We define such terms carefully, and justify numerically that the neglected terms are indeed numerically high order terms compared to the computable estimators. Numerical results for a variety of problems in 1D and 2D demonstrate that both the upper bound and lower bound are effective for measuring the error of eigenvalues and eigenfunctions.
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Robust recovery-type a posteriori error estimators for streamline upwind/Petrov Galerkin discretizations for singularly perturbed problems
Chongqing Jiaotong University（重庆交通大学）；Texas A&M International University（德州农工国际大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；Wayne State University（韦恩州立大学）
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Université de Lille 1（里尔第一大学）；INRIA Lille Nord Europe（法国国家信息与自动化研究所里尔北部分支）；Université de Valenciennes et du Hainaut Cambrésis（瓦朗谢纳及康布雷埃斯海诺特大学）
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