Abstract We study in this paper new developments of the Lagrange–Galerkin method for advection equation. In first part article we present a improved error estimate conventional method. second part, introduce local projection stabilized method, whereas third and analyze discontinuity-capturing Also, attention has been paid to influence quadrature rules on stability accuracy methods via numerical...

We consider space-time continuous Galerkin methods with mesh modification in time for semilinear second order hyperbolic equations. We show a priori estimates in the energy norm without mesh conditions. Under reasonable assumptions on the choice of the spatial mesh in each time step we show optimal order convergence rates. Estimates of the jump in the Riesz projection in two successive time ste...

In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term AXB − X + EFT = 0. These matrix equations appear in many applications in discrete-time control problems, filtering and image restoration and others. The proposed methods are based on projection onto the extended block Krylov subspace with a Galerkin approach (GA) or with the minimization of the...

In this paper we formulate a high order algorithm for hyperbolic conservation laws using a combination of Fourier expansions and unstructured spectral/hp elements. The unstructured part of the algorithm is formulated using a discontinuous Galerkin approximation. The conservation properties of the unstructured algorithm using a variable polynomial order are examined and an orthogonal projection ...

Various methods involving local proper orthogonal decomposition (POD) and Galerkin projection are presented aiming at accelerating the numerical integration of nonlinear, time dependent, dissipative problems. The approach combines short runs with a given computational fluid dynamics (CFD) solver and low dimensional models constructed by appropriate POD modes, which adapt to the dynamics. Applic...

In this paper, we study the semi-discrete Galerkin finite element method for parabolic equations with Lipschitz continuous coefficients. We prove the maximumnorm stability of the semigroup generated by the corresponding elliptic finite element operator, and prove the space-time stability of the parabolic projection onto the finite element space in L∞(QT ) and L p((0, T ); Lq ( )), 1 < p, q < ∞....

Let K be a completely continuous nonlinear integral operator, and consider solving x = K(x) by Galerkin's method. This can be written as xn = PnK(xn),Pn an orthogonal projection; the iterated Galerkin solution is defined by xn = K(xn). We give a general framework and error analysis for the numerical method that results from replacing all integrals in Galerkin's method with numerical integrals. ...

We design a numerical scheme for transport equations with oscillatory periodic scattering coefficients. The scheme is asymptotic preserving in the diffusion limit as Knudsen number goes to zero. It also captures the homogenization limit as the length scale of the scattering coefficient goes to zero. The proposed method is based on the construction of multiscale finite element basis and a Galerk...

In this paper, we discuss the superconvergence of the local discontinuous Galerkin methods for nonlinear convection-diffusion equations. We prove that the numerical solution is [Formula: see text]th-order superconvergent to a particular projection of the exact solution, when the upwind flux and the alternating fluxes are used. The proof is valid for arbitrary nonuniform regular meshes and for p...

We analyze the discretization of initial and boundary value problems with a stationary interface in one space dimension for the heat equation, the Schrödinger equation, and the wave equation by finite difference methods. Extending the concept of the elliptic projection, well known from the analysis of Galerkin finite element methods, to our finite difference case, we prove second-order error es...