Galerkin finite element methods for two-dimensional RLW and SRLW equations

In this paper, Galerkin finite methods for two-dimensional regularized long wave and symmetric regularized long wave equation are studied. The discretization in space is by Galerkin finite element method and in time is based on linearized backward Euler formula and extrapolated Crank–Nicolson scheme. Existence and uniqueness of the numerical solutions have been shown by Brouwer fixed point theorem. The error estimates of linearlized Crank–Nicolson method for RLW and SRLW equations are also presented. Numerical experiments, including the error norms and conservation variables, verify the efficiency and accuracy of the proposed numerical schemes.     

An efficient numerical method for singular perturbation problems

In this paper, we propose a method for the numerical solution of singularly perturbed two-point boundary-value problems (BVPs). First, we develop two schemes to integrate initial–value problem (IVP) for system of two first-order differential equations, and then by using these schemes we solve the BVP. Precisely, we convert the second-order BVP into a system of first-order differential equations, and then apply the numerical schemes to obtain the solution. In order to get an initial condition for the system, we use the asymptotic approximate solution. Error estimates are derived and numerical examples are provided to illustrate the present method.     

Wavelet-Taylor Galerkin Method for the Burgers Equation

In this paper, we propose a wavelet-Taylor Galerkin method for the numerical solution of the Burgers equation. In deriving the computational scheme, Taylor-generalized Euler time discretization is performed prior to wavelet-based Galerkin spatial approximation. The linear system of equations obtained in the process are solved by approximate-factorization-based simple explicit schemes, and the resulting solution is compared with that from regular methods. To deal with transient advection-diffusion situations that evolve toward a convective steady state, a splitting-up strategy is known to be very effective. So the Burgers equation is also solved by a splitting-up method using a wavelet-Taylor Galerkin approach. Here, the advection and diffusion terms in the Burgers equation are separated, and the solution is computed in two phases by appropriate wavelet-Taylor Galerkin schemes. Asymptotic stability of all the proposed schemes is verified, and the L_∞ errors relative to the analytical solution together with the numerical solution are reported. AMS subject classification (2000) 65M70     

On the convergence of the Galerkin method for coupled thermoelasticity problems

The Cauchy problem for a system of two operator-differential equations is considered that is an abstract statement of linear coupled thermoelasticity problems. Error estimates in the energy norm for the semidiscrete Galerkin method as applied to the Cauchy problem are established without imposing any special conditions on the projection subspaces. By way of illustration, the error estimates are applied to finite element schemes for solving the coupled problem of plate thermoelasticity considered within the framework of the Kirchhoff linearized theory. The results obtained are also applicable to the case when the projection subspaces in the Galerkin method (for the original abstract problem) are the eigenspaces of operators similar to unbounded self-adjoint positive definite operator coefficients of the original equations.     

A meshless local Galerkin method for the numerical solution of Hammerstein integral equations based on the moving least squares technique

In this paper, a computational scheme is proposed to estimate the solution of one- and two-dimensional Fredholm-Hammerstein integral equations of the second kind. The method approximates the solution using the discrete Galerkin method based on the moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The discrete Galerkin technique for integral equations results from the numerical integration of all integrals in the system corresponding to the Galerkin method. Since the proposed method is constructed on a set of scattered points, it does not require any background meshes and so we can call it as the meshless local discrete Galerkin method. The implication of the scheme for solving two-dimensional integral equations is independent of the geometry of the domain. The new method is simple, efficient and more flexible for most classes of nonlinear integral equations. The error analysis of the method is provided. The convergence accuracy of the new technique is tested over several Hammerstein integral equations and obtained results confirm the theoretical error estimates.     

A linearized Crank–Nicolson Galerkin FEMs for the nonlinear fractional Ginzburg–Landau equation

In this paper, we study a linearized Crank–Nicolson Galerkin finite element method for solving the nonlinear fractional Ginzburg–Landau equation. The boundedness, existence and uniqueness of the numerical solution are studied in details. Then we prove that the optimal error estimates hold unconditionally, in the sense that no restriction on the size of the time step in terms of the spatial mesh size needs to be assumed. Finally, numerical tests are investigated to support our theoretical analysis.     

FINITE ELEMENT NONLINEAR GALERKIN COUPLING METHOD FOR THE EXTERIOR STEADY NAVIER-STOKES PROBLEM

1.IntroductionNonlinearGalerkinmethodsaremultilevelschemesforthedissipativeevolutionpartialdifferentialequations.Theycorrespondtothesplittingsoftheunknownu:u=y z)wherethecomponentsareofdifferentorderofmagnitudewithrespecttoaparameterrelatedtothespati...     

A fully Galerkin method for the damped generalized regularized long‐wave (DGRLW) equation

In this article, a fully discrete Galerkin scheme based on a nonlinear Crank–Nicolson method to approximate the solution of the DGRLW equation is constructed. Some a priori bounds are proved as well as error estimates. Then, a linearized modification scheme by an extrapolation method is discussed. The two schemes are time second order convergence. The last part is devoted to some numerical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

非定常 N-S 方程的非线性 Galerkin 算法及其误差估计

本文给出了二维非定常N-S方程的三种数值格式,其中空间变量用谱非线性Galerkin算法进行离散,时间变量用有限差分离散,并研究了这些格式数值解的逼近精度．最后,给出了部分数值计算结果．     

Time-adaptive Adomian decomposition-based numerical scheme for Euler equations

Time efficiency is one of the more critical concerns in computational fluid dynamics simulations of industrial applications. Extensive research has been conducted to improve the underlying numerical schemes to achieve time process reduction. Within this context, this paper presents a new time discretization method based on the Adomian decomposition technique for Euler equations. The obtained scheme is time-order adaptive; the order is automatically adjusted at each time step and over the space domain, leading to significant processing time reduction. The scheme is formulated in an appropriate recursive formula, and its efficiency is demonstrated through numerical tests by comparison to exact solutions and the popular Runge–Kutta-discontinuous Galerkin method.     

A system of boundary integral equations for the transmission problem in acoustics

In this paper, we reduce the classical two-dimensional transmission problem in acoustic scattering to a system of coupled boundary integral equations (BIEs), and consider the weak formulation of the resulting equations. Uniqueness and existence results for the weak solution of corresponding variational equations are established. In contrast to the coupled system in Costabel and Stephan (1985) [4], we need to take into account exceptional frequencies to obtain the unique solvability. Boundary element methods (BEM) based on both the standard and a two-level fast multipole Galerkin schemes are employed to compute the solution of the variational equation. Numerical results are presented to verify the efficiency and accuracy of the numerical methods.     

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.     

Abstract theory of hybridizable discontinuous Galerkin methods for second-order quasilinear elliptic problems

An abstract theory for discretizations of second-order quasilinear elliptic problems based on the mixed-hybrid discontinuous Galerkin method. Discrete schemes are formulated in terms of approximations of the solution to the problem, its gradient, flux, and the trace of the solution on the interelement boundaries. Stability and optimal error estimates are obtained under minimal assumptions on the approximating space. It is shown that the schemes admit an efficient numerical implementation.     

Numerical and theoretical study of weak Galerkin finite element solutions of Turing patterns in reaction–diffusion systems

In this paper, we introduce numerical schemes and their analysis based on weak Galerkin finite element framework for solving 2‐D reaction–diffusion systems. Weak Galerkin finite element method (WGFEM) for partial differential equations relies on the concept of weak functions and weak gradients, in which differential operators are approximated by weak forms through the Green's theorem. This method allows the use of totally discontinuous functions in the approximation space. In the current work, the WGFEM solves reaction–diffusion systems to find unknown concentrations (u, v) in element interiors and boundaries in the weak Galerkin finite element space WG(P₀, P₀, RT₀) . The WGFEM is used to approximate the spatial variables and the time discretization is made by the backward Euler method. For reaction–diffusion systems, stability analysis and error bounds for semi‐discrete and fully discrete schemes are proved. Accuracy and efficiency of the proposed method successfully tested on several numerical examples and obtained results satisfy the well‐known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point. Acquired numerical results asserted the efficiency of the proposed scheme.     

A weak Galerkin finite element method for a coupled Stokes‐Darcy problem

In this article, we introduce and analyze a weak Galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes equations in primal velocity‐pressure formulation and Darcy equation in the second order primary formulation, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers‐Joseph‐Saffman law. By using the weak Galerkin approach, we consider the two‐dimensional problem with the piecewise constant elements for approximations of the velocity, pressure, and hydraulic head. Stability and optimal error estimates are obtained. Finally, we provide several numerical results illustrating the good performance of the proposed scheme and confirming the optimal order of convergence provided by the weak Galerkin approximation. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1352–1373, 2017     

Local discontinuous Galerkin approximations to fractional Bagley-Torvik equation

In this research, numerical approximation to fractional Bagley-Torvik equation as an important model arising in fluid mechanics is investigated. Our discretization algorithm is based on the local discontinuous Galerkin (LDG) schemes along with using the natural upwind fluxes, which enables us to solve the model problem element by element. This means that we require to solve a low-order system of equations in each subinterval, hence avoiding the need for a full global solution. The proposed schemes are tested on a range of initial- and boundary-value problems including a variable coefficient example, a nonsmooth problem, and some oscillatory test cases with exact solutions. Also, the validation of the proposed methods was compared with those obtained available existing computational procedures. Overall, it was found that LDG methods indicated highly satisfactory performance with comparatively lower degree of polynomials and number of elements compared with other numerical models.     

An optimal asymptotic-numerical method for convection dominated systems having exponential boundary layers

The convection dominated diffusion problems are studied. Higher order accurate numerical methods are presented for problems in one and two dimensions. The underlying technique utilizes a superposition of given problem into two independent problems. The first one is the reduced problem that refers to the outer or smooth solution. Stretching transformation is used to obtain the second problem for inner layer solution. The method considered for outer or degenerate problems are based on higher order Runge–Kutta methods and upwind finite differences. However, inner problem is solved analytically or asymptotically. The schemes presented are proved to be consistent and stable. Possible extensions to delay differential equations and to nonlinear problems are outlined. Numerical results for several test examples are illustrated and a comparative analysis is presented. It is observed that the method presented is highly accurate and easy to implement. Moreover, the numerical results obtained are not only comparable with the exact solution but also in agreement with the theoretical estimates.     

Hybridized schemes of the discontinuous Galerkin method for stationary convection–diffusion problems

For stationary linear convection–diffusion problems, we construct and study a new hybridized scheme of the discontinuous Galerkin method on the basis of an extended mixed statement of the problem. Discrete schemes can be used for the solution of equations degenerating in the leading part and are stated via approximations to the solution of the problem, its gradient, the flow, and the restriction of the solution to the boundaries of elements. For the spaces of finite elements, we represent minimal conditions responsible for the solvability, stability, accuracy, and superconvergence of the schemes. A new procedure for the post-processing of solutions of HDG-schemes is suggested.     

Interior penalty discontinuous Galerkin methods with implicit time‐integration techniques for nonlinear parabolic equations

We prove existence and numerical stability of numerical solutions of three fully discrete interior penalty discontinuous Galerkin methods for solving nonlinear parabolic equations. Under some appropriate regularity conditions, we give the l²(H¹) and l^∞(L²) error estimates of the fully discrete symmetric interior penalty discontinuous Galerkin–scheme with the implicit θ ‐schemes in time, which include backward Euler and Crank–Nicolson finite difference approximations. Our estimates are optimal with respect to the mesh size h. The theoretical results are confirmed by some numerical experiments. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

CONVERGENCE OF THE CRANK-NICOLSON/NEWTON SCHEME FOR NONLINEAR PARABOLIC PROBLEM

In this paper, the Crank-Nicolson/Newton scheme for solving numerically secondorder nonlinear parabolic problem is proposed. The standard Galerkin finite element method based on P2 conforming elements is used to the spatial discretization of the problem and the Crank-Nicolson/Newton scheme is applied to the time discretization of the resulted finite element equations. Moreover, assuming the appropriate regularity of the exact solution and the finite element solution, we obtain optimal error estimates of the fully discrete CrankNicolson/Newton scheme of nonlinear parabolic problem. Finally, numerical experiments are presented to show the efficient performance of the proposed scheme.