Abstract This paper aims to introduce an asymptotically exact a posteriori error estimator for non-selfadjoint Steklov eigenvalue problem arising from inverse scattering by using the complementary technique, which provides an asymptotically exact estimate for eigenpair of non-selfadjoint Steklov eigenvalue problem. Besides, as its applications, we design a novel cascadic adaptive method for non-selfadjoint Steklov eigenvalue problem based on the asymptotically exact estimate. In our novel algorithm, we will transform the non-selfadjoint Steklov eigenvalue problem into some boundary value problems on the adaptive space sequence and some non-selfadjoint Steklov eigenvalue problem on a low dimensional finite element space. The involved boundary value problems are solved by executing some smoothing steps which is the key point of cascadic algorithm. The mesh refinement strategy and the number of smoothing steps for the cascadic adaptive method will be controlled by the proposed asymptotically exact a posteriori error estimator.

中文翻译：

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非自伴Steklov特征值问题的渐近精确后验误差估计量

摘要 本文旨在利用互补技术为逆散射引起的非自伴Steklov特征值问题引入渐近精确后验误差估计器，为非自伴Steklov特征值问题的特征对提供渐近精确估计。此外，作为其应用，我们设计了一种新的基于渐近精确估计的非自伴随 Steklov 特征值问题的级联自适应方法。在我们的新算法中，我们将非自伴随 Steklov 特征值问题转化为自适应空间序列上的一些边值问题和低维有限元空间上的一些非自伴随 Steklov 特征值问题。涉及的边值问题是通过执行一些平滑步骤来解决的，这是级联算法的关键点。