Analysis and computational method based on quadratic B‐spline FEM for the Rosenau–Burgers equation

L_∞‐error estimates for B‐spline Galerkin finite element solution of the Rosenau–Burgers equation are considered. The semidiscrete B‐spline Galerkin scheme is studied using appropriate projections. For fully discrete B‐spline Galerkin scheme, we consider the Crank–Nicolson method and analyze the corresponding error estimates in time. Numerical experiments are given to demonstrate validity and order of accuracy of the proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 877–895, 2016     

Pointwise error estimates for a stabilized Galerkin method: non-selfadjoint problems

Highly localized pointwise error estimates for a stabilized Galerkin method are provided for second-order non-selfadjoint elliptic partial differential equations. The estimates show a local dependence of the error on the derivative of the solution u and weak dependence on the global norm. The results in this paper are an extension of the previous pointwise error estimates for the self-adjoint problems. In order to provide pointwise error estimates in the presence of the first-order term in the differential equations, we prove that the stabilized Galerkin solution is higher order perturbation to the Ritz projection of the true solutions. Then, we proceed to obtain pointwise estimates using the so-called discrete Green’s function. Application to error expansion inequalities and a posteriori error estimators are briefly discussed.     

三维非线性抛物型积分-微分方程的A.D.I.Galerkin方法及其分析

研究三维非线性抛物型积分-微分方程的A.D.I.Galerkin方法.通过交替方向,化三维为一维,简化计算;通过Galerkin法,保持高精度.成功处理了Volterra项的影响;对所提Galerkin及A.D.I.Galerkin格式给出稳定性和收敛性分析,得到最佳H1和L2模估计.     

Methods for the numerical solution of the Benjamin‐Bona‐Mahony‐Burgers equation

L^∞‐error estimates for finite element for Galerkin solutions for the Benjamin‐Bona‐Mahony‐Burgers (BBMB) equation are considered. A priori bound and the semidiscrete Galerkin scheme are studied using appropriate projections. For fully discrete Galerkin schemes, we consider the backward Euler method and analyze the corresponding error estimates. For a second order accuracy in time, we propose a three‐level backward method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A balanced atmospheric model of Lorenz

Error estimates are established in terms of the Rossby number for a nonlinear Galerkin approximation to the solutions of a reduced approximation to the balanced atmosphere model of Lorenz with mass forcing. Thereby the spectral dimension of the nonlinear Galerkin approximation is linked to the Rossby number and the diffusion coefficients.     

Meshless Galerkin algorithms for boundary integral equations with moving least square approximations

In this paper, we first give error estimates for the moving least square (MLS) approximation in the Hk norm in two dimensions when nodes and weight functions satisfy certain conditions. This two-dimensional error results can be applied to the surface of a three-dimensional domain. Then combining boundary integral equations (BIEs) and the MLS approximation, a meshless Galerkin algorithm, the Galerkin boundary node method (GBNM), is presented. The optimal asymptotic error estimates of the GBNM for three-dimensional BIEs are derived. Finally, taking the Dirichlet problem of Laplace equation as an example, we set up a framework for error estimates of the GBNM for boundary value problems in three dimensions.     

Variational formulation for the stationary fractional advection dispersion equation

In this article a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented. Appropriate fractional derivative spaces are defined and shown to be equivalent to the usual fractional dimension Sobolev spaces H^s. Existence and uniqueness results are proven, and error estimates for the Galerkin approximation derived. Numerical results are included that confirm the theoretical estimates. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Long- time Convergence of Numerical Approximations for Semilinear Parabolic Equations(Ⅰ )

§ 0 .Introduction　 We consider the numerical approximations of the dynamical systems governed bysemilinear parabolic equations,which are discretized by Galerkin and nonlinear Galerkinmethods in space,and by Runge-Kutta method in time.The numerical approximationson a finite time interval have already been widely studied(see[1 ]— [5] ) .We areconcerned with the long-time convergence and error estimates.This article is composedof three parts. In part ,we provide an abstract framework. In§…     

On best possible order of convergence estimates in the collocation method and Galerkin's method for singularly perturbed boundary value problems for systems of first-order ordinary differential equations

The collocation method and Galerkin method using parabolic splines are considered. Special adaptive meshes whose number of knots is independent of the small parameter of the problem are used. Unimprovable estimates in the -norm are obtained. For the Galerkin method these estimates are quasioptimal, while for the collocation method they are suboptimal.

     

Higher order convergence results for the Rayleigh-Ritz method applied to eigenvalue problems: 2. Improved error bounds for eigenfunctions

The application of the Rayleigh-Ritz method for approximating the eigenvalues and eigenfunctions of linear eigenvalue problems in several dimensions is investigated. The object is to improve upon known error estimates for the approximate eigenfunctions. Results for the Galerkin approximation of the eigenfunctions are developed under varying assumptions on the boundary conditions and domain of definition of the eigenvalue problem. These results, coupled with a previous result relating Galerkin and Rayleigh-Ritz approximation of the eigenfunctions, are then used to obtain improved error estimates for the approximate eigenfunctions in theL ² and uniform norms.This research was supported in part by AEC Grant (11-1)-2075.     

Discrete boundary element methods on general meshes in 3D

Abstract. This paper is concerned with the stability and convergence of fully discrete Galerkin methods for boundary integral equations on bounded piecewise smooth surfaces in . Our theory covers equations with very general operators, provided the associated weak form is bounded and elliptic on , for some . In contrast to other studies on this topic, we do not assume our meshes to be quasiuniform, and therefore the analysis admits locally refined meshes. To achieve such generality, standard inverse estimates for the quasiuniform case are replaced by appropriate generalised estimates which hold even in the locally refined case. Since the approximation of singular integrals on or near the diagonal of the Galerkin matrix has been well-analysed previously, this paper deals only with errors in the integration of the nearly singular and smooth Galerkin integrals which comprise the dominant part of the matrix. Our results show how accurate the quadrature rules must be in order that the resulting discrete Galerkin method enjoys the same stability properties and convergence rates as the true Galerkin method. Although this study considers only continuous piecewise linear basis functions on triangles, our approach is not restricted in principle to this case. As an example, the theory is applied here to conventional “triangle-based” quadrature rules which are commonly used in practice. A subsequent paper [14] introduces a new and much more efficient “node-based” approach and analyses it using the results of the present paper. Received December 10, 1997 / Revised version received May 26, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

Multidimensional characteristic Galerkin methods for hyperbolic systems

We consider characteristic Galerkin methods for the solution of hyperbolic systems of partial differential equations of first order. A new recipe for the construction of approximate evolution operators is given in order to derive consistent methods. With the help of semigroup theory we derive error estimates for classes of characteristic Galerkin methods. The theory is applied to the wave equation and also to the Euler equations of gas dynamics. In the latter case one can show that Fey's genuinely multidimensional method can be reinterpreted as a characteristic Galerkin method. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Analysis and finite element approximations for distributed optimal control problems for implicit parabolic equations

This work concerns analysis and error estimates for optimal control problems related to implicit parabolic equations. The minimization of the tracking functional subject to implicit parabolic equations is examined. Existence of an optimal solution is proved and an optimality system of equations is derived. Semi-discrete (in space) error estimates for the finite element approximations of the optimality system are presented. These estimates are symmetric and applicable for higher-order discretizations. Finally, fully-discrete error estimates of arbitrarily high-order are presented based on a discontinuous Galerkin (in time) and conforming (in space) scheme. Two examples related to the Lagrangian moving mesh Galerkin formulation for the convection-diffusion equation are described.     

Analysis of the parareal approach based on discontinuous Galerkin method for time-dependent Stokes equations

This paper analyzes a parareal approach based on discontinuous Galerkin (DG) method for the time-dependent Stokes equations. A class of primal discontinuous Galerkin methods, namely variations of interior penalty methods, are adopted for the spatial discretization in the parareal algorithm (we call it parareal DG algorithm). We study three discontinuous Galerkin methods for the time-dependent Stokes equations, and the optimal continuous in time error estimates for the velocities and pressure are derived. Based on these error estimates, the proposed parareal DG algorithm is proved to be unconditionally stable and bounded by the error of discontinuous Galerkin discretization after a finite number of iterations. Finally, some numerical experiments are conducted which confirm our theoretical results, meanwhile, the efficiency of the parareal DG algorithm can be seen through a parallel experiment.     

Navier-Stokes方程对称破坏分歧点的谱Galerkin逼近[英文]

本文研究了Navier-Stokes方程对称破坏分歧点的谱Galerkin逼近问题，构造了定常Navier-Stokes方程对称破坏分歧点扩充系统及其谱Galerkin逼近扩充系统，证明了谱Galerkin逼扩充系统解的存在性和收敛性，从而给出了Navier-Stokes方程对称破坏分歧点的谱Galerkin逼近，并给出了逼近的误差估计。     

L-error estimate for the strongly damped extensible beam equations

L²-error estimates for finite-element Galerkin solutions for the strongly damped extensible beam equations are considered. The semidiscrete scheme and a fully discrete Galerkin scheme based on the Crank-Nicolson method are studied using appropriate projections. The corresponding stability analysis and error estimates are obtained.     

Error analysis of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation

This article discusses a priori and a posteriori error estimates of discontinuous Galerkin finite element method for optimal control problem governed by the transport equation. We use variational discretization concept to discretize the control variable and discontinuous piecewise linear finite elements to approximate the state and costate variable. Based on the error estimates of discontinuous Galerkin finite element method for the transport equation, we get a priori and a posteriori error estimates for the transport equation optimal control problem. Finally, two numerical experiments are carried out to confirm the theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1493–1512, 2017     

Conditional ε-uniform boundedness of Galerkin projectors and convergence of an adaptive mesh method as applied to singularly perturbed boundary value problems

The Galerkin finite element method is applied to nonself-adjoint singularly perturbed boundary value problems on Shishkin meshes. The Galerkin projection method is used to obtain conditionally ε-uniform a priori error estimates and to prove the convergence of a sequence of meshes in the case of an unknown boundary layer edge.     

A fully discrete Galerkin method for Abel-type integral equations

In this paper, we present a Galerkin method for Abel-type integral equation with a general class of kernel. Stability and quasi-optimal convergence estimates are derived in fractional-order Sobolev norms. The fully-discrete Galerkin method is defined by employing simple tensor-Gauss quadrature. We develop a corresponding perturbation analysis which allows to keep the number of quadrature points small. Numerical experiments have been performed which illustrate the sharpness of the theoretical estimates and the sensitivity of the solution with respect to some parameters in the equation.     

Galerkin method for a third-order differential-operator equation

We study the Galerkin method for a third-order differential-operator equation with self-adjoint leading operator A and subordinate linear operator K(t) in a separable Hilbert space. We prove a theorem on the existence and uniqueness of a strong solution of the original problem. We derive estimates for the accuracy of the approximate solutions constructed by the Galerkin method. An application of the suggested method to the solution of a model problem is described.