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

In this paper, the Crank-Nicolson/Newton scheme for solving numerically second- order 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-Nieolson/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 Crank- Nicolson/Newton scheme of nonlinear parabolic problem. Finally, numerical experiments are presented to show the efficient performance of the proposed scheme.

Analysis of L1-Galerkin FEMs for Time-Fractional Nonlinear Parabolic Problems

This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality.In this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative.In terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems.The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.

TWO-GRID CHARACTERISTIC FINITE VOLUME METHODS FOR NONLINEAR PARABOLIC PROBLEMS＊

In this work, two-grid characteristic finite volume schemes for the nonlinear parabolic problem are considered. In our algorithms, the diffusion term is discretized by the finite volume method, while the temporal differentiation and advection terms are treated by the characteristic scheme. Under some conditions about the coefficients and exact solution, optimal error estimates for the numerical solution are obtained. Furthermore, the two- grid characteristic finite volume methods involve solving a nonlinear equation on coarse mesh with mesh size H, a large linear problem for the Oseen two-grid characteristic finite volume method on a fine mesh with mesh size h = O（H2） or a large linear problem for the Newton two-grid characteristic finite volume method on a fine mesh with mesh size h = 0（I log hll/2H3）. These methods we studied provide the same convergence rate as that of the characteristic finite volume method, which involves solving one large nonlinear problem on a fine mesh with mesh size h. Some numerical results are presented to demonstrate the efficiency of the proposed methods.

A MULTIGRID METHOD FOR NONLINEAR PARABOLIC PROBLEMS

The multigrid algorithm in [13] is developed for solving nonlinear parabolic equations arising from the finite element discretization. The computational cost of the algorithm is approximate O(NkN) where N-k is the dimension of the finite element space and N is the number of time steps.

MIXED FINITE ELEMENT METHODS FOR A STRONGLYNONLINEAR PARABOLIC PROBLEM

A mixed finite element method is developed to approximate the solution of a strongly nonlinear second-order parabolic problem. The existence and uniqueness of the approximation are demonstrated and L-2-error estimates are established for both the scalar function and the flux. Results are given for the continuous-time case.

LEAST-SQUARES MIXED FINITE ELEMENT METHODS FOR NONLINEAR PARABOLIC PROBLEMS

Focuses on the formulation of a number of least-squares mixed finite element schemes to solve the initial boundary value problem of a nonlinear parabolic partial differential equation. Convergence analysis; Information on the least-squares mixed element schemes for nonlinear parabolic problem.

INITIAL VALUE PROBLEM FOR A NONLINEAR PARABOLIC EQUATION WITH SINGULAR INTEGRAL-DIFFERENTIAL TERM

We study the initial value problem for a nonlinear parabolic equation with singular integral-differential term. By means of a series of a priori estimations of the solutions to the problem andLeray-Schauder fixed point principle, we demonstrate the existence and uniqueness theorems ofthe generalized and classical global solutions. Lastly, we discuss the asymptotic properties of thesolution as t tends to infinity.

ERROR ESTIMATES FOR TWO-SCALE COMPOSITE FINITE ELEMENT APPROXIMATIONS OF NONLINEAR PARABOLIC EQUATIONS

We study spatially semidiscrete and fully discrete two-scale composite nite element method for approximations of the nonlinear parabolic equations with homogeneous Dirichlet boundary conditions in a convex polygonal domain in the plane.This new class of nite elements,which is called composite nite elements,was rst introduced by Hackbusch and Sauter[Numer.Math.,75(1997),pp.447-472]for the approximation of partial di erential equations on domains with complicated geometry.The aim of this paper is to introduce an effcient numerical method which gives a lower dimensional approach for solving partial di erential equations by domain discretization method.The composite nite element method introduces two-scale grid for discretization of the domain,the coarse-scale and the ne-scale grid with the degrees of freedom lies on the coarse-scale grid only.While the ne-scale grid is used to resolve the Dirichlet boundary condition,the dimension of the nite element space depends only on the coarse-scale grid.As a consequence,the resulting linear system will have a fewer number of unknowns.A continuous,piecewise linear composite nite element space is employed for the space discretization whereas the time discretization is based on both the backward Euler and the Crank-Nicolson methods.We have derived the error estimates in the L^(∞)(L^(2))-norm for both semidiscrete and fully discrete schemes.Moreover,numerical simulations show that the proposed method is an efficient method to provide a good approximate solution.