Given a linear self-adjoint differential operator \(\mathscr {L}\) along with a discretization scheme (like Finite Differences, Finite Elements, Galerkin Isogeometric Analysis, etc.), in many numerical applications it is crucial to understand how good the (relative) approximation of the whole spectrum of the discretized operator \(\mathscr {L}\,^{(n)}\) is, compared to the spectrum of the continuous operator \(\mathscr {L}\). The theory of Generalized Locally Toeplitz sequences allows to compute the spectral symbol function \(\omega \) associated to the discrete matrix \(\mathscr {L}\,^{(n)}\). Inspired by a recent work by T. J. R. Hughes and coauthors, we prove that the symbol \(\omega \) can measure, asymptotically, the maximum spectral relative error \(\mathscr {E}\ge 0\). It measures how the scheme is far from a good relative approximation of the whole spectrum of \(\mathscr {L}\), and it suggests a suitable (possibly non-uniform) grid such that, if coupled to an increasing refinement of the order of accuracy of the scheme, guarantees \(\mathscr {E}=0\).

给定线性自伴随微分算子\(\mathscr {L}\)以及离散化方案（如有限差分、有限元、伽辽金等几何分析等），在许多数值应用中，了解与连续算子\(\mathscr {L}\)的频谱相比，离散化算子\(\mathscr {L}\,^{(n)}\)的整个频谱的（相对）近似值。广义局部托普利兹序列的理论允许计算与离散矩阵\(\mathscr {L}\,^{(n)}\)关联的谱符号函数\(\omega \)。受 TJR Hughes 及其合著者最近一项工作的启发，我们证明了符号\(\omega \)可以渐近地测量最大谱相对误差\(\mathscr {E}\ge 0\)。它衡量了该方案如何远离\(\mathscr {L}\)的整个频谱的良好相对近似，并且它建议了一个合适的（可能是非均匀的）网格，这样，如果耦合到对该方案的准确性顺序，保证\(\mathscr {E}=0\)。