In a projection-based model reduction, Galerkin-type projection is frequently used to generate reduced matrix. However, the stability may not be preserved and computational effort for generating negligible. this study, we use random projections reduce an original large-scale We show that of matrix guaranteed with high probability can obtained efficiently.

We present de-aliasing rules to be used when evaluating non-linear terms with polynomial spectral methods on nonuniform grids analogous to the de-aliasing rules used in Fourier spectral methods. They are based upon the idea of super-collocation followed by a Galerkin projection of the non-linear terms. We demonstrate through numerical simulation that both accuracy and stability can be greatly e...

In this paper, a symplectic model reduction technique, proper symplectic decomposition (PSD) with symplectic Galerkin projection, is proposed to save the computational cost for the simplification of large-scale Hamiltonian systems while preserving the symplectic structure. As an analogy to the classical proper orthogonal decomposition (POD)-Galerkin approach, PSD is designed to build a symplect...

How close are Galerkin eigenvectors to the best approximation available out of the trial subspace ? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace – and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors. Both orthogonal...

A local proper orthogonal decomposition (POD) plus Galerkin projection method is applied to the unsteady lid-driven cavity problem, namely the incompressible fluid flow in a twodimensional box whose upper wall is moved back and forth at moderately large values of the Reynolds number. Such a method was recently developed for one-dimensional parabolic problems. Its extension to fluid dynamics pro...

How close are Galerkin eigenvectors to the best approximation available out of the trial subspace? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace—and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors. Both orthogonal-Ga...

In this paper, we give the first a priori error analysis of the hybridizable discontinuous Galerkin (HDG) methods for Timoshenko beams. The analysis is based on the use of a projection especially designed to fit the structure of the numerical traces of the HDG method. This property allows to prove in a very concise manner that the projection of the errors is bounded in terms of the distance bet...

We deal with the numerical solution of time-dependent convection-diffusion-reaction equations. We combine the local projection stabilization method for the space discretization with two different time discretization schemes: the continuous Galerkin-Petrov (cGP) method and the discontinuous Galerkin (dG) method of polynomial of degree k. We establish the optimal error estimates and present numer...

In uncertainty quantification, critical parameters of mathematical models are substituted by random variables. We consider dynamical systems composed of ordinary differential equations. The unknown solution is expanded into an orthogonal basis of the random space, e.g., the polynomial chaos expansions. A Galerkin method yields a numerical solution of the stochastic model. In the linear case, th...