Limit Galerkin-Petrov schemes for a nonlinear convection-diffusion equation

In the present paper, we suggest a version of the nonconformal finite-element method (a perturbed Galerkin method) for approximating a quasilinear convection-diffusion equation in divergence form. A grid scheme is constructed with the use of an approach based on the Galerkin-Petrov approximation to the mixed statement of the original problem. The separated coordinate approximation of the solution components for the mixed problem permits one to take into account the direction of convective transport and preserve the main properties of the spatial operator of the original problem. We prove the stability of the line method scheme and a two-layer weighted scheme for the original problem.     

Jacobi Spectral Methods for Volterra-Urysohn Integral Equations of Second Kind with Weakly Singular Kernels

Abstract

In this article, we discuss Jacobi spectral Galerkin and iterated Jacobi spectral Galerkin methods for Volterra-Urysohn integral equations with weakly singular kernels and obtain the convergence results in both the infinity and weighted L²-norm. We show that the order of convergence in iterated Jacobi spectral Galerkin method improves over Jacobi spectral Galerkin method. We obtain the convergence results in two cases when the exact solution is sufficiently smooth and non-smooth. For finding the improved convergence results, we also discuss Jacobi spectral multi-Galerkin and iterated Jacobi spectral multi-Galerkin method and obtain the convergence results in weighted L²-norm. In fact, we prove that the iterated Jacobi spectral multi-Galerkin method improves over iterated Jacobi spectral Galerkin method. We provide numerical results to verify the theoretical results.     

Nonconformal schemes of the finite-element method for nonlinear hyperbolic conservation laws

We suggest a method for constructing grid schemes for initial-boundary value problems for many-dimensional nonlinear systems of first-order equations of hyperbolic type on the basis of the Galerkin-Petrov limit approximation to the mixed statement of an original problem. Our grid schemes are versions of the nonconformal finite-element method in which the approximate solution is constructed in the space of piecewise polynomial functions that admit discontinuities on the boundary of triangulation elements of the design domain.     

On the convergence of solutions for a class of nonconformal FEM schemes for quasilinear elliptic equations

We consider versions of the nonconformal finite element method for the approximation to a second-order quasilinear elliptic equation in divergence form. For the construction of grid schemes, we use an approach used earlier for the nonstationary convection-diffusion equation and based on the Galerkin-Petrov limit approximation to the mixed statement of the original problem. The accuracy of solutions of nonconformal schemes with triangular linear finite elements is estimated in the absence of interior penalty terms, which are usually used in methods close to DG-methods for the stabilization of the scheme solution.     

GLOBAL FINITE ELEMENT NONLINEAR GALERKIN METHOD FOR THE PENALIZED NAVIER-STOKES EQUATIONS GLOBAL FINITE ELEMENT NONLINEAR GALERKIN METHOD FOR THE PENALIZED NAVIER-STOKES EQUATIONS

1. IntroductionIn the numerical simulation of the Navier-Stokes equations one encounters three seriousdifficulties in the case of large Reynolds numbers f the treatment of the incomPressibility con-dition divu = 0, the treatment of the noIilinear terms and the large time integration. For thetreatment of the incoInPressibility condition, one use the penalty method in the case of finiteelemellts [1--2l and for the treatmen of the noulinar terms and the large tfor integration, oneuse the nonlin…     

Galerkin-petrov limit schemes for the convection-diffusion equation

In the present paper, we suggest a method for constructing grid schemes for the multidimensional convection-diffusion equation. The method is based on the approximation of the integral identity that is used in the definition of a weak solution of the differential problem. The use of spaces of smooth trial functions and spaces of functions with possible discontinuities in which the solution of the original problem is sought naturally leads to Galerkin-Petrov methods. The suggested method for the construction of grid schemes is based on a finite-element semidiscretization of the original space with respect to space variables, which constructs the space of trial functions on the basis of the direction of the convective transport near the boundaries of finite elements, the limit passage from a scheme with smooth trial functions to schemes with discontinuous trial functions, and the further discretization of the resulting equations with respect to the time variable. We prove the stability of the constructed difference schemes and present the results of computations for model problems.     

On the stability of the spectral Galerkin approximation

We study stability properties of the spectral Galerkin approximation of the solutions of semilinear problems. Assuming that the data of the problem are known within a certain error, we investigate when the solution of the Galerkin approximate equation provides a desired accuracy uniformly with respect to small perturbations of the data. We show that for certain classes of semilinear problems an additional compactness assumption is sufficient to assure that the spectral Galerkin method provides an accurate approximation to the exact solution uniformly with respect to small perturbations of the data. This revised version was published online in June 2006 with corrections to the Cover Date.     

NONLINEAR GALERKIN METHOD FOR NAVIER-STOKES EQUATIONS WITH STREAM-VORTICITY FORM

I Let D be a launder domain in Rz with Lipehitz continuous bodare I'. ac consider the stream-vortidty form of hmedependent Navier-stokes Muahons describing the 'flow Of a ~ incompreSSible nuid confined in Dwhere to and & are vorticity and stream function. BecaUSe the action (1) doeS not include the differentialten z, ~ tie ~ con&tion of' dab at In tthe Paper, we give a ho element n~ Galerkin ~, acs ~ is ~ an tab finite element spaceS X. and X* for the aPPmxhaation of the ~ ac v~ fUncti…     

Comparison of the finite difference and Galerkin methods as applied to the solution of the hydrodynamic equations

The three-dimensional nonlinear hydrodynamic equations which describe wind induced flow in a homogeneous sea are transformed from Cartesian coordinates into sigma coordinates. The solution of these equations in the horizontal is accomplished using a standard finite difference grid and established finite difference methods.The accuracy and computational efficiency, in terms of both computer time and main memory requirements, of using either the Galerkin method or a finite difference grid through the vertical is considered. Calculations, using the same number of functions in the Galerkin method as grid bases through the vertical shows that the Galerkin method has superior accuracy over the grid box method. Hence, for a given accuracy a smaller number of functions than grid boxes may be used, with associated saving in computational resources.For the case in which the vertical variation of eddy viscosity is fixed, an eigenvalue problem can be solved to yield a set of eigenfunctions. Using these eigenfunctions as a basis set with the Galerkin approach, a Galerkin-eigenfunction method is developed. Calculations show that the Galerkin-eigenfunction technique is accurate and in a linear model is clearly computationally more economic than the use of grid boxes through the vertical.     

Analysis of Wave Propagation in 1D Inhomogeneous Media

In this paper, we consider the one-dimensional inhomogeneous wave equation with particular focus on its spectral asymptotic properties and its numerical resolution. In the first part of the paper, we analyze the asymptotic nodal point distribution of high-frequency eigenfunctions, which, in turn, gives further information about the asymptotic behavior of eigenvalues and eigenfunctions. We then turn to the behavior of eigenfunctions in the high- and low-frequency limit. In the latter case, we derive a homogenization limit, whereas in the first we show that a sort of self-homogenization occurs at high frequencies. We also remark on the structure of the solution operator and its relation to desired properties of any numerical approximation. We subsequently shift our focus to the latter and present a Galerkin scheme based on a spectral integral representation of the propagator in combination with Gaussian quadrature in the spectral variable with a frequency-dependent measure. The proposed scheme yields accurate resolution of both high- and low-frequency components of the solution and as a result proves to be more accurate than available schemes at large time steps for both smooth and nonsmooth speeds of propagation.     

Numerical analysis of a modified finite element nonlinear Galerkin method

Summary. A fully discrete modified finite element nonlinear Galerkin method is presented for the two-dimensional equation of Navier-Stokes type. The spatial discretization is based on two finite element spaces X_H and X_h defined on a coarse grid with grid size H and a fine grid with grid size h << H, respectively; the time discretization is based on the Euler explicit scheme with respect to the nonlinear term. We analyze the stability and convergence rate of the method. Comparing with the standard finite element Galerkin method and the nonlinear Galerkin method, this method can admit a larger time step under the same convergence rate of same order. Hence this method can save a large amount of computational time. Finally, we provide some numerical tests on this method, the standard finite element Galerkin method, and the nonlinear Galerkin method, which are in a good agreement with the theoretical analysis.Mathematics Subject Classification (2000): 35Q30, 65M60, 65N30, 76D05     

On Exact Results in the Finite Element Method

We prove that the finite element method for one-dimensional problems yields no discretization error at nodal points provided the shape functions are appropriately chosen. Then we consider a biharmonic problem with mixed boundary conditions and the weak solution u. We show that the Galerkin approximation of u based on the so-called biharmonic finite elements is independent of the values of u in the interior of any subelement.     

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms.     

Stability and convergence of optimum spectral non‐linear Galerkin methods

Our objective in this article is to present some numerical schemes for the approximation of the 2‐D Navier–Stokes equations with periodic boundary conditions, and to study the stability and convergence of the schemes. Spatial discretization can be performed by either the spectral Galerkin method or the optimum spectral non‐linear Galerkin method; time discretization is done by the Euler scheme and a two‐step scheme. Our results show that under the same convergence rate the optimum spectral non‐linear Galerkin method is superior to the usual Galerkin methods. Finally, numerical example is provided and supports our results. Copyright © 2001 John Wiley & Sons, Ltd.     

An optimal nonlinear Galerkin method with mixed finite elements for the steady Navier‐Stokes equations

An optimal nonlinear Galerkin method with mixed finite elements is developed for solving the two‐dimensional steady incompressible Navier‐Stokes equations. This method is based on two finite element spaces X_H and X_h for the approximation of velocity, defined on a coarse grid with grid size H and a fine grid with grid size h ? H, respectively, and a finite element space M_h for the approximation of pressure. We prove that the difference in appropriate norms between the solutions of the nonlinear Galerkin method and a classical Galerkin method is of the order of H⁵. If we choose H = O(h^2/5), these two methods have a convergence rate of the same order. We numerically demonstrate that the optimal nonlinear Galerkin method is efficient and can save a large amount of computational time. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 762–775, 2003.     

A postprocessed Galerkin method with Chebyshev or Legendre polynomials

Summary. We present an approximate-inertial-manifold-based postprocess to enhance Chebyshev or Legendre spectral Galerkin methods. We prove that the postprocess improves the order of convergence of the Galerkin solution, yielding the same accuracy as the nonlinear Galerkin method. Numerical experiments show that the new method is computationally more efficient than Galerkin and nonlinear Galerkin methods. New approximation results for Chebyshev polynomials are presented. Received January 5, 1998 / Revised version received September 7, 1999 / Published online June 8, 2000     

Convergence and stability of finite element nonlinear Galerkin method for the Navier-Stokes equations

In this article, we present a new fully discrete finite element nonlinear Galerkin method, which are well suited to the long time integration of the Navier-Stokes equations. Spatial discretization is based on two-grid finite element technique; time discretization is based on Euler explicit scheme with variable time step size. Moreover, we analyse the boundedness, convergence and stability condition of the finite element nonlinear Galerkin method. Our discussion shows that the time step constraints of the method depend only on the coarse grid parameter and the time step constraints of the finite element Galerkin method depend on the fine grid parameter under the same convergence accuracy. Received February 2, 1994 / Revised version received December 6, 1996     

Homogenization and a galerkin approximation in three–dimensional beam theory

In this work we study the limit relations between a Galerkin approximation method, of the spectral type, for the three–dimensional linearized elasticity model of a multicellular beam, and the respective homogenization process. We show that the choice of the basis functions, of the Galerkin approximation, .commutes with the homogenization process but the same does not hold for the final approximations of the three-dimensional solution     

Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations

Summary. A nonlinear Galerkin method using mixed finite elements is presented for the two-dimensional incompressible Navier-Stokes equations. The scheme is based on two finite element spaces and for the approximation of the velocity, defined respectively on one coarse grid with grid size and one fine grid with grid size and one finite element space for the approximation of the pressure. Nonlinearity and time dependence are both treated on the coarse space. We prove that the difference between the new nonlinear Galerkin method and the standard Galerkin solution is of the order of $H^2$, both in velocity ( and pressure norm). We also discuss a penalized version of our algorithm which enjoys similar properties. Received October 5, 1993 / Revised version received November 29, 1993     

Spectral Galerkin method and its application to a Cauchy problem of Helmholtz equation

Based on a mathematical model of laser beams, we present a spectral Galerkin method for solving a Cauchy problem of the Helmholtz equation in a rectangle, where the Cauchy data pairs are given at y?=?0 and boundary data are for x?=?0 and x?=?π. The solution is sought in the interval 0?<?y?<?1. The spectral Galerkin method is considered as a regularization method. We then perform an analysis on the error bound for this method. For illustration, several numerical experiments are constructed to demonstrate the feasibility and efficiency of the proposed method.