双调和算子多重和聚集特征值的\(C^0\)IPG 逼近的后验误差估计器
An a Posteriori Error Estimator for $C^0$ IPG Approximation of Multiple and Clustered Eigenvalues of the Biharmonic Operator
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中文总结 AI 辅助
基于\(C^0\)IPG 离散化，针对双调和算子多重和聚集特征值相关特征函数提出后验误差估计器，经聚类投影分析及引入辅助估计器，使可靠性和效率界具鲁棒性，数值实验验证了理论结果及估计器的鲁棒性。
中文摘要 AI 辅助
本文基于\(C^0\)内部罚有限元（\(C^0\)IPG）离散化，针对双调和算子多重和聚集特征值相关的特征函数提出并分析了一种后验误差估计器。该估计器能有效捕捉目标特征函数的局部奇异性，在自适应过程中起重要作用。通过基于聚类投影的严格分析并引入辅助理论误差估计器，将不变子空间误差与可计算残差估计器联系起来。所得的可靠性和效率界在网格大小、网格级别及目标聚类内的内部谱隙方面具有鲁棒性。数值实验支持理论结果并证明了该估计器的鲁棒性。
英文摘要
In this paper, based on the $C^0$ interior penalty Galerkin ($C^0$IPG) discretization, we propose and analyze an a posteriori error estimator for eigenfunctions associated with multiple and clustered eigenvalues of the biharmonic operator. The proposed estimator may capture the local singularities of the target eigenfunctions efficiently and play an important role in adaptive procedures. We develop a rigorous cluster-projection-based analysis and introduce an auxiliary theoretical error estimator to connect the invariant subspace error with a computable residual estimator. Most importantly, the resulting reliability and efficiency bounds are robust with respect to the mesh size, the mesh level and the internal spectral gaps within the target cluster. Numerical experiments support the theoretical results and demonstrate the robustness of the proposed estimator.
