1,081 research outputs found
On stability of discretizations of the Helmholtz equation (extended version)
We review the stability properties of several discretizations of the
Helmholtz equation at large wavenumbers. For a model problem in a polygon, a
complete -explicit stability (including -explicit stability of the
continuous problem) and convergence theory for high order finite element
methods is developed. In particular, quasi-optimality is shown for a fixed
number of degrees of freedom per wavelength if the mesh size and the
approximation order are selected such that is sufficiently small and
, and, additionally, appropriate mesh refinement is used near
the vertices. We also review the stability properties of two classes of
numerical schemes that use piecewise solutions of the homogeneous Helmholtz
equation, namely, Least Squares methods and Discontinuous Galerkin (DG)
methods. The latter includes the Ultra Weak Variational Formulation
Relative perturbation theory: IV. sin 2θ theorems
AbstractThe double angle theorems of Davis and Kahan bound the change in an invariant subspace when a Hermitian matrix A is subject to an additive perturbation A→Ã=A+ΔA. This paper supplies analogous results when A is subject to a congruential, or multiplicative, perturbation A→Ã=D*AD. The relative gaps that appear in the bounds involve the spectrum of only one matrix, either A or Ã, in contrast to the gaps that appear in the single angle bounds.The double angle theorems do not directly bound the difference between the old invariant subspace S and the new one S̃ but instead bound the difference between S̃ and its reflection JS̃ where the mirror is S and J reverses S⊥, the orthogonal complement of S. The double angle bounds are proportional to the departure from the identity and from orthogonality of the matrix D̃=defD−1JDJ. Note that D̃ is invariant under the transformation D→D/αforα≠0, whereas the single angle theorems give bounds proportional to D's departure from the identity and from orthogonality.The corresponding results for the singular value problem when a (nonsquare) matrix B is perturbed to B̃=D*1BD2 are also presented
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