A fully discrete nonlinear Galerkin method for the 3D Navier–Stokes equations

The purpose of this paper is twofold: (i) We show that the Fourier‐based Nonlinear Galerkin Method (NLGM) constructs suitable weak solutions to the periodic Navier–Stokes equations in three space dimensions provided the large scale/small scale cutoff is appropriately chosen. (ii) If smoothness is assumed, NLGM always outperforms the Galerkin method by a factor equal to 1 in the convergence order of the H ¹‐norm for the velocity and the L²‐norm for the pressure. This is a purely linear superconvergence effect resulting from standard elliptic regularity and holds independently of the nature of the boundary conditions (whether periodicity or no‐slip BC is enforced). © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A discrete Galerkin method for a hyperslngular boundary integral equation

Received on 14 August 1995. Revised on 20 August 1996. Consider solving the interior Neumann problem with D a simply-connected planar region and S=D a smooth curve.A double-layer potential is used to represent the solution,and it leads to the problem of solving a hypersingular integralequation. This integral equation is reformulated as a Cauchysingular integral equation. A discrete Galerkin method withtrigonometric polynomials is then given for its solution. Anerror analysis is given, and numerical examples complete thepaper.     

A method for the numerical resolution of Abel-type integral equations of the first kind

A numerical method to solve Abel-type integral equations of first kind is given. In this paper we suggest the research of a numerical solution for Abel-type integral equations of the first kind, by using a collocation method employing an interpolatory product-quadrature formula with a trigonometric polynomial of the first order. Some results of numerical examples are reported.     

Fully discrete Galerkin methods for systems of boundary integral equations

In this paper we analyze a family of full discretizations of spline Galerkin methods for a class of systems of boundary integral equations of the first kind with logarithmic principal part. We prove the existence of an asymptotic expansion of the error of the Galerkin and the optimal order Galerkin collocation method. We finally derive asymptotic expansions for some common postprocessings of the solutions, both exactly and under the effect of additional discretization. Some examples where these techniques apply are provided.     

Stability and robustness of collocation methods for Abel-type integral equations

Summary We study stability aspects of collocation methods for Abel-type integral equations of the first kind using piecewise polynomials. These collocation methods may be formulated as projection methods. Stability is defined as the boundedness of the sequence of projectors in their natural setting. Robustness is essentially the optimal asymptotic insensitivity to perturbations in the data. We show that stability and robustness are equivalent for the above collocation methods. This allows us to obtain optimal error estimates for some methods that are well-known to be robust. We also present numerical results for some methods which appear to be robust.Research supported in part by the United States Army under Contract No. DAAG 29-83-K-0109     

Legendre spectral Galerkin method for second-kind Volterra integral equations

The Legendre spectral Galerkin method for the Volterra integral equations of the second kind is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the L ₂ norm) will decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical examples are given to illustrate the theoretical results.      

Richardson extrapolation of iterated discrete Galerkin solution for two-dimensional Fredholm integral equations

In this paper, we study the numerical solution of two-dimensional Fredholm integral equation by discrete Galerkin and iterated discrete Galerkin method. We are able to derive an asymptotic error expansion of the iterated discrete Galerkin solution. This expansion covers arbitrarily high powers of the discretization parameters if the solution of the integral equation is smooth. The expansion gives rise to Richardson-type extrapolation schemes which rapidly improve the original rate of the convergence. Numerical experiments confirm our theoretical results.     

The discrete collocation method for nonlinear integral equations

The collocation method for solving linear and nonlinear integralequations results in many integrals which must be evaluatednumerically. In this paper, we give a general framework fordiscrete collocation methods, in which all integrals are replacedby numerical integrals. In some cases, the collocation methodleads to solutions which are superconvergent at the collocationnode points. We consider generalizations of these results, toobtain similar results for discrete collocation solutions. Lastly,we consider a variant due to Kumar and Sloan for the collocationsolution of Hammerstein integral equations.     

Stability and convergence of a fully discrete local discontinuous Galerkin method for multi-term time fractional diffusion equations

In this paper, a fully discrete local discontinuous Galerkin method for a class of multi-term time fractional diffusion equations is proposed and analyzed. Using local discontinuous Galerkin method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established. By choosing the numerical flux carefully, we prove that the method is unconditionally stable and convergent with order O(h ^k+1 + (Δt)^2?α ), where k, h, and Δt are the degree of piecewise polynomial, the space, and time step sizes, respectively. Numerical examples are carried out to illustrate the effectiveness of the numerical scheme.     

A trigonometric Galerkin method for volume integral equations arising in TM grating scattering

Transverse magnetic (TM) scattering of an electromagnetic wave from a periodic dielectric diffraction grating can mathematically be described by a volume integral equation.This volume integral equation, however, in general fails to feature a weakly singular integral operator. Nevertheless, after a suitable periodization, the involved integral operator can be efficiently evaluated on trigonometric polynomials using the fast Fourier transform (FFT) and iterative methods can be used to solve the integral equation. Using Fredholm theory, we prove that a trigonometric Galerkin discretization applied to the periodized integral equation converges with optimal order to the solution of the scattering problem. The main advantage of this FFT-based discretization scheme is that the resulting numerical method is particularly easy to implement, avoiding for instance the need to evaluate quasiperiodic Green’s functions.     

A fully discrete stabilized finite element method for the time-dependent Navier-Stokes equations

In this article, we consider a fully discrete stabilized finite element method based on two local Gauss integrations for the two-dimensional time-dependent Navier-Stokes equations. It focuses on the lowest equal-order velocity-pressure pairs. Unlike the other stabilized method, the present approach does not require specification of a stabilization parameter or calculation of higher-order derivatives, and always leads to a symmetric linear system. The Euler semi-implicit scheme is used for the time discretization. It is shown that the proposed fully discrete stabilized finite element method results in the optimal order bounds for the velocity and pressure.     

Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems

In this paper, we propose and analyze a fully discrete local discontinuous Galerkin (LDG) finite element method for time-fractional fourth-order problems. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. Stability is ensured by a careful choice of interface numerical fluxes. We prove that our scheme is unconditional stable and convergent. Numerical examples are shown to illustrate the efficiency and accuracy of our scheme.     

Sparse multiscale representation of Galerkin method for solving linear-mixed Volterra-Fredholm integral equations

This paper presents an efficient method for solving the linear-mixed Volterra-Fredholm equations using multiscale transformation. For this purpose, by changing the variables, the Fredholm-Volterra equation is discretized using wavelet Galerkin method. This equation reduces to a set of linear algebraic equations by using the wavelet transform matrix and the operational matrix of integration. To reach the sparse coefficients matrix for having a reduction in the computational cost, thresholding is used. This sparse system solves by generalized minimal residual (GMRES) method. If the appropriate threshold selects, the number of nonzero coefficients reduces while the error will not be less than a certain amount. The convergence analysis has been investigated. The validity and applicability of the technique are illustrated by a series of numerical tests.     

Two-grid stabilized mixed finite element method for fully discrete reaction-diffusion equations

Two-grid mixed finite element method is proposed based on backward Euler schemes for the unsteady reaction-diffusion equations. The scheme combines with the stabilized mixed finite element scheme by using the lowest equal-order pairs for the velocity and pressure. The space two-grid method is also used to reduce the time consuming. The benefits of this approach are to avoid the higher derivative, but to have more favorable stability, and to get the numerical solution of the two unknown variables simultaneously. Stability analysis and error estimates are given in this work. Finally, the theoretical results are verified by the numerical examples.     

Dissipative/conservative Galerkin method using discrete partial derivatives for nonlinear evolution equations

A new method is proposed for designing Galerkin schemes that retain the energy dissipation or conservation properties of nonlinear evolution equations such as the Cahn–Hilliard equation, the Korteweg–de Vries equation, or the nonlinear Schrödinger equation. In particular, as a special case, dissipative or conservative finite-element schemes can be derived. The key device there is the new concept of discrete partial derivatives. As examples of the application of the present method, dissipative or conservative Galerkin schemes are presented for the three equations with some numerical experiments.     

On the convergence of the Galerkin method for nonsmooth solutions of integral equations

Summary It is shown that the stability region of the Galerkin method includes solutions not lying in the conventional energy space. Optimal order error estimates for these nonsmooth solutions are derived. The new result is compared with the classical statement by means of the basic potential problem.     

A Galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions

We consider retarded boundary integral formulations of the three-dimensional wave equation in unbounded domains. Our goal is to apply a Galerkin method in space and time in order to solve these problems numerically. In this approach the computation of the system matrix entries is the major bottleneck. We will propose new types of finite-dimensional spaces for the time discretization. They allow variable time-stepping, variable order of approximation and simplify the quadrature problem arising in the generation of the system matrix substantially. The reason is that the basis functions of these spaces are globally smooth and compactly supported. In order to perform numerical tests concerning our new basis functions we consider the special case that the boundary of the scattering problem is the unit sphere. In this case explicit solutions of the problem are available which will serve as reference solutions for the numerical experiments.     

A note on the convergence of product integration and Galerkin method for weakly singular integral equations

We consider weakly singular integral equations of Fredholm-type whose kernels satisfy certain algebraic estimates with their derivatives. In particular, we establish optimal convergence order estimates for product integration and Galerkin method applied on suitable grading mesh for the solution of such equations. Some superconvergence results are also derived.