Implicit Boundary Conditions in Partial Differential Equations Discretizations: Identifying Spurious Modes and Model Reduction

TL;DR

This paper investigates how improper handling of boundary conditions in PDE discretizations causes spurious modes, proposing new quality tests to identify and reduce these inaccuracies without prior eigenvalue knowledge.

Contribution

It introduces a standardized boundary condition treatment and new quality metrics to detect and eliminate spurious modes in PDE eigenvalue computations.

Findings

Implicit boundary conditions are often violated in discretizations, leading to spurious modes.

Approximately half of the computed spectrum in many problems is of low quality.

The proposed tests effectively identify low-accuracy modes for model reduction.

Abstract

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.