We revisit the problem of spurious modes that are sometimes encountered in partial differential equations discretizations. It is generally suspected that one of the causes for spurious modes is due to how boundary conditions are treated, and we use this as the starting point of our investigations. By regarding boundary conditions as algebraic constraints on a differential equation, we point out that any differential equation with homogeneous boundary conditions also admits a typically infinite number of hidden or implicit boundary conditions. In most discretization schemes, these additional implicit boundary conditions are violated, and we argue that this is what leads to the emergence of spurious modes. These observations motivate two definitions of the quality of computed eigenvalues based on violations of derivatives of boundary conditions on the one hand, and on the Grassmann distance between subspaces associated with computed eigenspaces on the other. Both of these tests are based on a standardized treatment of boundary conditions and do not require a priori knowledge of eigenvalue locations. The effectiveness of these tests is demonstrated on several examples known to have spurious modes. In addition, these quality tests show that in most problems, about half the computed spectrum of a differential operator is of low quality. The tests also specifically identify the low accuracy modes, which can then be projected out as a type of model reduction scheme.

翻译：我们重新审视了偏微分方程离散化中偶尔出现的伪模态问题。普遍认为，边界条件的处理方式是导致伪模态的原因之一，并将此作为研究的出发点。通过将边界条件视为微分方程的代数约束，我们指出任何具有齐次边界条件的微分方程也通常隐含着无限数量的隐藏或隐式边界条件。在大多数离散化方案中，这些额外的隐式边界条件被违反，我们认为这正是导致伪模态出现的原因。这些观察结果促使我们基于两个标准来定义计算特征值的质量：一方面基于边界条件导数的违反程度，另一方面基于与计算特征空间相关联的子空间之间的Grassmann距离。这两种测试都基于边界条件的标准化处理，且不需要预先知道特征值的位置。我们通过多个已知存在伪模态的算例验证了这些测试的有效性。此外，这些质量测试表明，在大多数问题中，微分算子计算谱的大约一半是低质量的。这些测试还能具体识别出低精度模态，然后可以将其作为一类模型降阶方案投影剔除。