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首页> 外文期刊>SIAM Journal on Numerical Analysis >Boundary quasi-orthogonality and sharp inclusion bounds for large dirichlet eigenvalues
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Boundary quasi-orthogonality and sharp inclusion bounds for large dirichlet eigenvalues

机译：大狄利克雷特征值的边界拟正交性和尖锐包含域

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摘要

We study eigenfunctions fj and eigenvalues φj of the Dirichlet Laplacian on a bounded domain Ω C ?~n with piecewise smooth boundary. We bound the distance between an arbitrary parameter E > 0 and the spectrum {Ej} in terms of the boundary L~2-norm of a normalized trial solution u of the Helmholtz equation (δ + E)u = 0. We also bound the L~2-norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all E greater than a small constant, and improve upon the best-known bounds of Moler-Payne by a factor of the wavenumber vE. One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions. In the case of planar, strictly star-shaped domains we give an inclusion bound where the constant is also sharp. We give explicit constants in the theorems, and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy. The proof makes use of a new quasi-orthogonality property of the boundary normal derivatives of the eigenmodes (Theorem 1.3), of interest in its own right. Namely, the operator norm of the sum of rank 1 operators δnφj [δnφj over all Ej in a spectral window of width √E-a sum with about E(n-1)/2 terms-is at most a constant factor (independent of E) larger than the operator norm of any one individual term.
机译：我们研究具有分段光滑边界的有界域ΩC？〜n上的Dirichlet拉普拉斯算子的特征函数fj和特征值φj。根据Helmholtz方程（δ+ E）u = 0的归一化试验解u的边界L〜2-范数，我们将任意参数E> 0与频谱{Ej}的距离限制。本征函数误差的本征函数的L〜2范数。这两个结果都是尖锐的常数，对于所有E都大于一个小常数，并且以波数vE改善了最著名的Moler-Payne边界。一种应用是通过例如特定解决方案的方法来高频地解决特征值问题。在平面的严格星形区域中，我们给出了一个常数也很尖锐的包含边界。我们在定理中给出显式常数，并显示一个数值示例，其中将2500th附近的特征值计算为14位相对精度。该证明利用了本征模的边界法线导数的新的准正交性质（定理1.3），这本身就是令人感兴趣的。即，在具有约E（n-1）/ 2项的宽度√Ea和的频谱窗口中，所有Ej上的1级算子δnφj[δnφj的总和的算子范数最多为一个常数（与E无关）大于任何一个单独术语的运算符规范。

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