Error Estimates for Mixed Finite Element Methods for Sobolev Equation

The purpose of this paper is to investigate the convergence of the mixed finite element method for the initial-boundary value problem for the Sobolev equation Ut-div{aut + b1 u} = f based on the Raviart-Thomas space Vh × Wh H(div; × L2(). Optimal order estimates are obtained for the approximation of u, ut, the associated velocity p and divp respectively in L(0,T;L2()), L(0,T;L2()), L(0,T;L2()2), and L2(0, T; L2()). Quasi-optimal order estimates are obtained for the approximations of u, ut in L(0, T; L()) and p in L(0,T; L()2).

A reduced-order extrapolation algorithm based on CNLSMFE formulation and POD technique for two-dimensional Sobolev equations

A reduced-order extrapolation algorithm based on Crank-Nicolson least-squares mixed finite element （CNLSMFE） formulation and proper orthogonal decomposition （POD） technique for two-dimensional （2D） Sobolev equations is established. The error estimates of the reduced-order CNLSMFE solutions and the implementation for the reduced-order extrapolation algorithm are provided. A numerical example is used to show that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order extrapolation algorithm is feasible and efficient for seeking numerical solutions to 2D Sobolev equations.

Anisotropic rectangular nonconforming finite element analysis for Sobolev equations

An anisotropic rectangular nonconforming finite element method for solving the Sobolev equations is discussed under semi-discrete and full discrete schemes. The corresponding optimal convergence error estimates and superclose property are derived, which are the same as the traditional conforming finite elements. Furthermore, the global superconvergence is obtained using a post-processing technique. The numerical results show the validity of the theoretical analysis.

Two new least-squares mixed finite element procedures for convection-dominated Sobolev equations

Two new convection-dominated are derived under the approximate solutions least-squares mixed finite element procedures are formulated for solving Sobolev equations. Optimal H（div;Ω）×H1（Ω） norms error estimates standard mixed finite spaces. Moreover, these two schemes provide the with first-order and second-order accuracy in time increment, respectively.

A new mixed scheme based on variation of constants for Sobolev equation with nonlinear convection term

A new mixed scheme which combines the variation of constants and the H1-Galerkin mixed finite element method is constructed for nonlinear Sobolev equation with nonlinear con- vection term. Optimal error estimates are derived for both semidiscrete and fully discrete schemes. Finally, some numerical results are given to confirm the theoretical analysis of the proposed method.

AN ANISOTROPIC NONCONFORMING FINITE ELEMENT METHOD FOR APPROXIMATING A CLASS OF NONLINEAR SOBOLEV EQUATIONS

An anisotropic nonconforming finite element method is presented for a class of nonlinear Sobolev equations. The optimal error estimates and supercloseness are obtained for both semi-discrete and fully-discrete approximate schemes, which are the same as the traditional finite element methods. In addition, the global superconvergence is derived through the postprocessing technique. Numerical experiments are included to illustrate the feasibility of the proposed method.

Superconvergence Analysis and Extrapolation of Quasi-Wilson Nonconforming Finite Element Method for Nonlinear Sobolev Equations

Quasi-Wilson nonconforming finite element approximation for a class of nonlinear Sobolev equa- tions is discussed on rectangular meshes. We first prove that this element has two special characters by novel approaches. One is that （Vh（U -- Ihu）,VhVh）h may be estimated as order O（h2） when u E H3（Ω）, where Iuu denotes the bilinear interpolation of u, vh is a polynomial belongs to quasi-Wilson finite element space and △h denotes the piecewise defined gradient operator, h is the mesh size tending to zero. The other is that the consistency error of this element is of order O（h2）/O（h3） in broken Hi-norm, which is one/two order higher than its interpolation error when u ε Ha（Ω）/H4 （（1）. Then we derive the optimal order error estimate and su- perclose property via mean-value method and the known high accuracy result of bilinear element. Furthermore, we deduce the global superconvergence through interpolation post processing technique. At last, an extrapola- tion result of order O（h3）, two order higher than traditional error estimate, is obtained by constructing a new suitable extrapolation scheme.

Nonconforming H^1-Galerkin Mixed FEM for Sobolev Equations on Anisotropic Meshes

A nonconforming H^1-Calerkin mixed finite element method is analyzed for Sobolev equations on anisotropic meshes. The error estimates are obtained without using Ritz-Volterra projection.

A MODIFIED WEAK GALERKIN FINITE ELEMENT METHOD FOR SOBOLEV EQUATION

For Sobolev equation, we present a new numerical scheme based on a modified weak Galerkin finite element method, in which differential operators are approximated by weak forms through the usual integration by parts. In particular, the numerical method allows the use of discontinuous finite element functions and arbitrary shape of element. Optimal order error estimates in discrete H^1 and L^2 norms are established for the corresponding modified weak Galerkin finite element solutions. Finally, some numerical results are given to verify theoretical results.

STREAMLINE DIFFUSION F.E.M. FOR SOBOLEV EQUATIONS WITH CONVECTION DOMINATED TERM

In this paper,a streamline diffusion F.E.M. for linear Sobolev equations with convection dominated term is given.According to the range of space time F.E mesh parameter h ,two choices for artifical diffusion parameter δ are presented,and for the corresponding computation schemes the stability and error estimates in suitable norms are estabilished.

Time discontinuous Galerkin space-time finite element method for nonlinear Sobolev equations

This article presents a complete discretization of a nonlinear Sobolev equation using space-time discontinuous Galerkin method that is discontinuous in time and continuous in space. The scheme is formulated by introducing the equivalent integral equation of the primal equation. The proposed scheme does not explicitly include the jump terms in time, which represent the discontinuity characteristics of approximate solution. And then the complexity of the theoretical analysis is reduced. The existence and uniqueness of the approximate solution and the stability of the scheme are proved. The optimalorder error estimates in L2 （H1） and L2 （L2） norms are derived. These estimates are valid under weak restrictions on the space-time mesh, namely, without the condition kn ≥ ch2, which is necessary in traditional space-time discontinuous Galerkin methods. Numerical experiments are presented to verify the theoretical results.

A NEW CHARACTERISTIC EXPANDED MIXED METHOD FOR SOBOLEV EQUATION WITH CONVECTION TERM

In this paper,a new numerical method based on a new expanded mixed scheme and the characteristic method is developed and discussed for Sobolev equation with convection term.The hyperbolic part d(x)∂u/∂t+c(x,t)·∇u is handled by the characteristic method and the diffusion term∇·(a(x,t)∇u+b(x,t)∇ut)is approximated by the new expanded mixed method,whose gradient belongs to the simple square integrable(L^(2)(Ω))^(2)space instead of the classical H(div;Ω)space.For a priori error estimates,some important lemmas based on the new expanded mixed projection are introduced.An optimal priori error estimates in L^(2)-norm for the scalar unknown u and a priori error estimates in(L^(2))^(2)-norm for its gradientλ,and its fluxσ(the coefficients times the negative gradient)are derived.In particular,an optimal priori error estimate in H1-norm for the scalar unknown u is obtained.

FINITE ELEMENT METHODS FOR SOBOLEV EQUATIONS

Presents a study which formulated a new high-order time-stepping finite element method based upon the high-order numerical integration formula for Sobolev equations. Derivation of the optimal and superconvergence error estimates; Error estimates of convergence and superconvergence for the time-continuous finite element method; Details of the global superconvergence for the semi-discrete scheme.

INFINITELY MANY SOLUTIONS FOR A NONLINEAR ELLIPTIC EQUATION INVOLVING CRITICAL SOBOLEV EXPONENT

In this paper, it is proved that the following boundary value problem [GRAPHICS] admits infinitely many solution for 0 < lambda < lambda-1, n greater-than-or-equal-to 5 and for ball regions OMEGA = B(R)(0).

SOLUTIONS OF THE BENJAMIN-ONO EQUATION IN FRACTIONAL ORDER SOBOLEV SPACES

We show for the Benjamin-Ono equation an existence uniqeness theorem in Sobolev spaces of arbitrary fractional order s greater-than-or-equal-to 2, provided the initial data is given in the same space.

NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE

In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt ＋ umx ＋ 2ux m = 0, m =（1- δx^2）^2u,u（x, 0） = u0（x） ∈ H^s（R）, x ∈ R, t 〉 0,and show that the solution map is not uniformly continuous in Sobolev spaces H^s（R） for s 〉 7/2. Compared with the periodic problem, the non-periodic problem is more difficult,e.g., it depends on the conservation law. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part.

A PRIORI ESTIMATES TO THE MAXIMUM MODULUS OF GENERALIZED SOLUTIONS OF A CLASS OF QUASILINEAR ELLIPTIC EQUATIONS WITH ANISOTROPIC GROWTH CONDITIONS

In this paper we give a priori estimates for the maximum modulus of generalizedsolulions of the quasilinear elliplic equations irith anisotropic growth condition.