A collocation-Galerkin method is defined for Poisson's equation on the unit 2 square, using tensor products of continuous piecewise polynomials. Optimal L and Hq orders of convergence are established. This procedure requires fewer quadratures than the corresponding Galerkin procedure. Introduction. The collocation-Galerkin method was first introduced by Diaz [2], [3] for the two-point boundary ...

In this paper, we present a Hybridizable Discontinuous Galerkin (HDG) method for the solution of the compressible Euler and Navier-Stokes equations. The method is devised by using the discontinuous Galerkin approximation with a special choice of the numerical fluxes and weakly imposing the continuity of the normal component of the numerical fluxes across the element interfaces. This allows the ...

In this article we study the properties of the hyperinterpolation operator on the unit disk D in R. We show how the hyperinterpolation can be used in connection with the Kumar-Sloan method to approximate the solution of a nonlinear Poisson equation on the unit disk (discrete Galerkin method). A bound for the norm of the hyperinterpolation operator in the space C(D) is derived. Our results prove...

In this paper, we consider the Legendre spectral Galerkin and Legendre spectral collocation methods to approximate the solution of Urysohn integral equation. We prove that the approximated solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the same orders, O(n−r) in L2-norm and O(n 1 2 −r) in infinity norm, and the iterated Legendre Galerkin ...

The development of mathematical models for describing the dynamic behaviours of fluid conveying pipes, micro-pipes and nanotubes under the influence of some thermo-mechanical parameters results into nonlinear equations that are very difficult to solve analytically. In cases where the exact analytical solutions are presented either in implicit or explicit forms, high skills and rigorous mathemat...

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...

This paper is aimed at analyzing the existence and convergence of approximate solutions in shape optimization. Two questions arise when one applies a Ritz-Galerkin discretization to solve the necessary condition: does there exists an approximate solution and how good does it approximate the solution of the original infinite dimensional problem? We motivate a general setting by some illustrative...

We present a generalization of the finite volume evolution Galerkin scheme [17, 18] for hyperbolic systems with spatially varying flux functions. Our goal is to develop a genuinely multi-dimensional numerical scheme for wave propagation problems in a heterogeneous media. We illustrate our methodology for acoustic waves in a heterogeneous medium but the results can be generalized to more complex...

We present a high-order hybridizable discontinuous Galerkin method for solving elliptic interface problems in which the solution and gradient are nonsmooth because of jump conditions across the interface. The hybridizable discontinuous Galerkin method is endowed with several distinct characteristics. First, they reduce the globally coupled unknowns to the approximate trace of the solution on el...

In a recent paper we have introduced a postprocessing procedure for the Galerkin method for dissipative evolution partial differential equations with periodic boundary conditions. The postprocessing technique uses approximate inertial manifolds to approximate the high modes (the small scale components) in the exact solutions in terms of the Galerkin approximations, which in this case play the r...