N. Anders Petersson

Heinz-Otto Kreiss, N. Anders Petersson

We study the half-plane problem for the elastic wave equation subject to a free surface boundary condition, with particular emphasis on almost incompressible materials. A normal mode analysis is developed to estimate the solution in terms of the boundary data, showing that the problem is boundary stable. The dependence on the material properties, which is difficult to analyze by the energy method, is made transparent by our estimates. The normal mode technique is used to analyze the influence of truncation errors in a finite difference approximation. Our analysis explains why the number of grid points per wave length must be increased when the shear modulus ($μ$) becomes small, that is, for almost incompressible materials. To obtain a fixed error in the phase velocity of Rayleigh surface waves as $μ\to 0$, our analysis predicts that the grid size must be proportional to $μ^{1/2}$ for a second order method. For a fourth order method, the grid size can be proportional to $μ^{1/4}$. Numerical experiments confirm this scaling and illustrate the superior efficiency of the fourth order method.

Stefanie Günther, N. Anders Petersson

We consider the problem of recovering a unitary eigendecomposition of a complex unitary matrix from that of its embedded real-valued formulation. Such formulations arise naturally in scientific computing workflows that employ real-arithmetic solvers by representing complex matrices in term of their real and imaginary parts. While the reconstruction is trivial when the spectrum of the real-valued embedding is simple, degenerate and/or complex conjugated eigenvalues introduce ambiguities because each eigenspace may include contributions from both the unitary matrix and its complex conjugate. We prove that this ambiguity can always be resolved by applying a structured projection to the eigenspaces of the real-valued embedding, followed by a rank-revealing orthonormalization. The resulting procedure recovers the eigenvalues and an unitary eigenbasis for the original unitary matrix, with correct multiplicities of degenerate eigenvalues.