We derive error bounds for the Rayleigh-Ritz method for the approximation to extremal eigenpairs of a symmetric matrix. The bounds are expressed in terms of the eigenvalues of the matrix and the angle between the subspace and the eigenvector. We also present a sharp bound.

The convergence analysis of the bilinear finite element method to a class of non-linear degenerate wave equation on anisotropic meshes is considered in this paper. Moreover, the global superconvergence for semidiscrete scheme is proposed through interpolation instead of the Ritz Volterra projection of the exact solution.

After giving a panoramic view of the " textbook " interpretation of the new quantum mechanics, as a sequel to the old quantum theory, the conceptual basis of quantum theory since the Copenhagen Interpretation is reviewed in the context of various proposals since Einstein and Niels Bohr, designed to throw light on possible new facets bearing on its foundations, the key issues to its inherent " i...

Least-squares principles use artificial " energy " functionals to provide a Rayleigh-Ritz-like setting for the finite element method. These function-als are defined in terms of PDE's residuals and are not unique. We show that viable methods result from reconciliation of a mathematical setting dictated by the norm-equivalence of least-squares functionals with practicality constraints dictated by...

We consider the stability and convergence analysis of pressure stabilized finite element approximations of the transient Stokes’ equation. The analysis is valid for a class of symmetric pressure stabilization operators, but also for standard, inf-sup stable, velocity/pressure spaces without stabilization. Provided the initial data is chosen as a specific (method dependent) Ritz-projection, we g...

We study the Lanczos method for computing extreme eigenvalues of a symmetric or Hermitian matrix. It is not guaranteed that the extreme Ritz values are close to the extreme eigenvalues—even when the norms of the corresponding residual vectors are small. Assuming that the starting vector has been chosen randomly, we compute probabilistic bounds for the extreme eigenvalues from data available dur...

We present a Ritz-Galerkin discretization on sparse grids using pre-wavelets, which allows to solve elliptic differential equations with variable coefficients for dimension d = 2, 3 and higher dimensions d > 3. The method applies multilinear finite elements. We introduce an efficient algorithm for matrix vector multiplication using a Ritz-Galerkin discretization and semi-orthogonality. This alg...