The asymptotic rate of convergence of the Legendre–Galerkin spectral approximation to an atmospheric acoustic eigenvalue problem is established, as the dimension of the approximating subspace approaches infinity. Convergence is in the [Formula: see text] Sobolev norm and is based on the existing theory [F. Chatelin, Spectral Approximations of Linear Operators (SIAM, 2011)]. The assumption is made that the eigenvalues are simple. Numerical results that help interpret the theory are presented. Eigenvalues corresponding to acoustic modes with smaller [Formula: see text] norms are especially accurately approximated, even with lower dimensioned basis sets of Legendre polynomials. The deficiencies in the potential applications of the theoretical results are noted in connection with the numerical examples.

中文翻译：

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构造大气声正常模式的勒让德-伽辽金谱法的收敛性

当逼近子空间的维数接近无穷大时，Legendre-Galerkin 谱逼近对大气声学特征值问题的渐近收敛速度被建立。收敛在 [公式：见正文] Sobolev 范数中，并且基于现有理论 [F. Chatelin，线性算子的光谱近似（SIAM，2011）]。假设特征值很简单。给出了有助于解释该理论的数值结果。与具有较小 [公式：见文本] 范数的声学模式相对应的特征值特别准确地近似，即使是勒让德多项式的维数较低的基组。结合数值例子指出了理论结果在潜在应用方面的不足。