We present an overview of hybridizable discontinuous Galerkin methods for solving partial differential equations (PDEs) in continuum mechanics [1, 2, 3, 4]. The essential ingredients are a local Galerkin projection of the underlying PDEs at the element level onto spaces of polynomials of degree k to parametrize the numerical solution in terms of the numerical trace; a judicious choice of the nu...

We present a generalized polynomial chaos method to solve the steady and unsteady heat transfer problems with uncertainty in boundary conditions, diffusivity coefficient and forcing terms. The stochastic inputs and outputs are represented spectrally by employing the orthogonal polynomial functionals from the Askey scheme, as a generalization of the original polynomial chaos idea of Wiener [1]. ...

An iterative method for the low-rank approximate solution of a class of generalized Lyapunov equations is studied. At each iteration, a standard Lyapunov is solved using Galerkin projection with an extended Krylov subspace method. This Lyapunov equation is solved inexactly, thus producing a nonstationary iteration. Several theoretical and computational issues are discussed so as to make the ite...

Nonlinear model reduction is a critical component in the design process of control strategies for complex flows in the nonlinear regime. A Galerkin projection based on POD-modes is developed in this study that takes advantage of multiple bases for particular components of the governing equations. These bases are then linked by the Discrete Empirical Interpolation Method (DEIM) to ultimately arr...

Stability analysis of spatially discretized approximations to invariant circle solutions to a noninvertible map is shown to produce seemingly spurious eigenvalues, found to exist for all truncation numbers. Numerical analysis of the spectral Galerkin projection procedure used to compute the solutions and the eigenvalues of the linearization shows that the set of discrete eigenvalues converges i...

We propose and analyze a new hybridizable discontinuous Galerkin (HDG) method for second-order elliptic problems. Our method is obtained by inserting the L-orthogonal projection onto the approximate space for a numerical trace into all facet integrals in the usual HDG formulation. The orders of convergence for all variables are optimal if we use polynomials of degree k + l, k + 1 and k, where k...

This paper concerns the estimation of apparent motion from an image sequence. A model, named full model, is defined to describe the dynamics of motion field and images. The Proper Orthogonal Decomposition (POD) is used to derive basis to characterize motion fields and images. The Galerkin projection of the full model on these basis defines the so-called reduced model. Coefficients of motion fie...

A visualization of three-dimensional incompressible flows by divergence-free quasi-twodimensional projections of the velocity field onto three coordinate planes is revisited. An alternative and more general way to compute the projections is proposed. The approach is based on the Chorin projection combined with a SIMPLE-like iteration. Compared to the previous methodology based on divergence-fre...

We investigate nonlinear dynamical models for selfsustained oscillations in the flow past a rectangular cavity. The models are based on the method of Proper Orthogonal Decomposition (POD) and Galerkin projection, and we introduce an inner product and formulation of the equations of motion which enables one to use vector-valued POD modes for compressible flows. We obtain models between 3 and 20 ...

A family of orthogonal systems of irrational functions on the semi-infinite interval is introduced. The proposed orthogonal systems are based on Jacobi polynomials through an irrational coordinate transform. This family of orthogonal systems offers great flexibility to match a wide range of asymptotic behaviors at infinity. Approximation errors by the basic orthogonal projection and various oth...