One application of harmonic Ritz values is to approximate, with a projection method, the interior eigenvalues of a matrix A while avoiding the explicit use of the inverse A. In this context, harmonic Ritz values are commonly derived from a Petrov-Galerkin condition for the residual of a vector from the test space. In this paper, we investigate harmonic Ritz values from a slightly different pers...

Interior a priori error estimates in Sobolev norms are derived from interior RitzGalerkin equations which are common to a class of methods used in approximating solutions of second order elliptic boundary value problems. The estimates are valid for a large class of piecewise polynomial subspaces used in practice, which are defined on both uniform and nonuniform meshes. It is shown that the erro...

Let V be a Hilbert space, a(·, ·) : V × V → lR a bounded, V-elliptic bilinear form and : V → lR a bounded linear functional. We want to approximate the variational equation: Find u ∈ V such that (3.1) a(u, v) = (v) , v ∈ V. We recall that the Lax-Milgram Lemma ensures the existence and uniqueness of a solution of (3.1). Now, given a finite-dimensional subspace V h ⊂ V , dim V h = n h , the Ritz...

Linear multistep methods are considered which have a stability region S and are D-stable on the whole boundary S c S of S. Error estimates are derived which hold uniformly for the class of initial value problems Y’ AY + B(t), t > 0, Y(0) Y with normal matrix A satisfying the spectral condition Sp(AtA) S At O time step, Sp(A) spectrum of A. Because of this property, the result can be applied to ...

For a given subspace, the q-Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the q-Rayleigh-Ritz method defines the q-Ritz values and the q-Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the...