A POSTERIORI ERROR ESTIMATES OF FINITEELEMENT METHOD FOR PARABOLIC PROBLEMS

This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of parabolic problems in two space dimensions. The analysis extends previous results for elliptic problems to the parabolic context.

A SINGLE STEP SCHEME WITH HIGH ACCURACY FOR PARABOLIC PROBLEM

A single step scheme with high accuracy for solving parabolic problem is proposed. It is shown that this scheme possesses good stability and fourth order accuracy with respect to both time and space variables, which are superconvergent.

Entropy Unilateral Solution for Some Noncoercive Nonlinear Parabolic Problems Via a Sequence of Penalized Equations

We give an existence result of the obstacle parabolic equations3b（x,u） div（a（x,t,u, Vu））＋div（（x,t,u））=f in QT, 3twhere b（x,u） is bounded function ot u, the term atva,x,r,u, v u）） is a Letay type operator and the function is a nonlinear lower order and satisfy only the growth condition. The second term belongs to L1 （QT）. The proof of an existence solution is based on the penalization methods.

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.

Hlder Stability Estimate for an Inverse Parabolic Problem

This paper deals with a parabolic system in a multi dimentional bounded domain ΩR n with the smooth boundary Ω. We discuss an inverse parabolic problem of determining the indirectly measurable internal heat distribution at any intermediate moment from the heat distribution measurements in arbitrary accessible subdomain ωΩ at some time interval. Our main result is the Hlder stability estimate in the inverse problem and the proof is completed with a Carleman estimate and a eigenfunction expansion for parabolic equations.

A Quadratic Finite Volume Method for Parabolic Problems

.In this paper,a quadratic finite volume method(FVM)for parabolic problems is studied.We first discretize the spatial variables using a quadratic FVM to obtain a semi-discrete scheme.We then employ the backward Euler method and the Crank-Nicolson method respectively to further disctetize the time vatiable so as to derive two full-discrete schemes.The existence and uniqueness of the semi-discrete and full-discrete FVM solutions are established and their optimal error estimates are derived.Finally,we give numerical examples to illustrate the theoretical results.

A Stable FE-FD Method for Anisotropic Parabolic PDEs with Moving Interfaces

In this paper,a new finite element and finite difference(FE-FD)method has been developed for anisotropic parabolic interface problems with a known moving interface using Cartesian meshes.In the spatial discretization,the standard P,FE discretization is applied so that the part of the coefficient matrix is symmetric positive definite,while near the interface,the maximum principle preserving immersed interface discretization is applied.In the time discretization,a modified Crank-Nicolson discretization is employed so that the hybrid FE-FD is stable and second order accurate.Correction terms are needed when the interface crosses grid lines.The moving interface is represented by the zero level set of a Lipschitz continuous function.Numerical experiments presented in this paper confirm second orderconvergence.

CASCADIC MULTIGRID FOR PARABOLIC PROBLEMS CASCADIC MULTIGRID FOR PARABOLIC PROBLEMS

In this paper, we develop the cascadic multigrid method for parabolic problems. The optimal convergence accuracy and computation complexity are obtained.

Cascadic multigrid methods for parabolic problems

In this paper,we consider the cascadic multigrid method for a parabolic type equation.Backward Euler approximation in time and linear finite element approximation in space are employed.A stability result is established under some conditions on the smoother.Using new and sharper estimates for the smoothers that reflect the precise dependence on the time step and the spatial mesh parameter,these conditions are verified for a number of popular smoothers.Optimal error bound sare derived for both smooth and non-smooth data.Iteration strategies guaranteeing both the optimal accuracy and the optimal complexity are presented.

MULTIGRID FOR THE MORTAR FINITE ELEMENT FOR PARABOLIC PROBLEM

In this paper, a mortar finite element method for parabolic problem is presented. Multi-grid method is used for solving the resulting discrete system. It is shown that the multigrid method is optimal, i.e, the convergence rate is independent of the mesh size L and the time step parameter τ.

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.

A LOW ORDER NONCONFORMING ANISOTROPIC FINITE ELEMENT APPROXIMATION TO PARABOLIC PROBLEM

A low order nonconforming finite element is applied to the parabolic problem with anisotropicmeshes.Both the semidiscrete and fully discrete forms are studied.Some superclose properties andsuperconvergence are obtained through some novel approaches and techniques.

ON L2 ERROR ESTIMATE FOR WEAK GALERKIN FINITE ELEMENT METHODS FOR PARABOLIC PROBLEMS

A weak Galerkin finite element method with stabilization term, which is symmetric, positive definite and parameter free, was proposed to solve parabolic equations by using weakly defined gradient operators over discontinuous functions. In this paper, we derive the optimal order error estimate in L2 norm based on dual argument. Numerical experiment is conducted to confirm the theoretical results.

TIME-EXTRAPOLATION ALGORITHM （TEA） FOR LINEAR PARABOLIC PROBLEMS

The fast solutions of Crank-Nicolson scheme on quasi-uniform mesh for parabolic prob- lems are discussed. First, to decrease regularity requirements of solutions, some new error estimates are proved. Second, we analyze the two characteristics of parabolic discrete scheme, and find that the efficiency of Multigrid Method （MG） is greatly reduced. Nu- merical experiments compare the efficiency of Direct Conjugate Gradient Method （DCG） and Extrapolation Cascadic Multigrid Method （EXCMG）. Last, we propose a Time- Extrapolation Algorithm （TEA）, which takes a linear combination of previous several level solutions as good initial values to accelerate the rate of convergence. Some typical extrapolation formulas are compared numerically. And we find that under certain accuracy requirement, the CG iteration count for the 3-order and 7-level extrapolation formula is about 1/3 of that of DCG＇s. Since the TEA algorithm is independent of the space dimension, it is still valid for quasi-uniform meshes. As only the finest grid is needed, the proposed method is regarded very effective for nonlinear parabolic problems.

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.

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.

The Blow-up Rate for Positive Solutions of Indefinite Parabolic Problems and Related Liouville Type Theorems

In this paper, we derive an upper bound estimate of the blow-up rate for positive solutions of indefinite parabolic equations from Liouville type theorems. We also use moving plane method to prove the related Liouville type theorems for semilinear parabolic problems.

The Existence of Entropy Solutions to Some Parabolic Problems with L^1 Data

In this paper, an existence result of entropy solutions to some parabolic problems is established. The data belongs to L^1 and no growth assumption is made on the lower-order term in divergence form.

UNIFORM QUADRATIC CONVERGENCE OF A MONOTONE WEIGHTED AVERAGE METHOD FOR SEMILINEAR SINGULARLY PERTURBED PARABOLIC PROBLEMS＊

This paper deals with a monotone weighted average iterative method for solving semilinear singularly perturbed parabolic problems. Monotone sequences, based on the ac- celerated monotone iterative method, are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. An analysis of uniform convergence of the monotone weighted average iterative method to the solutions of the nonlinear difference scheme and continuous problem is given. Numerical experiments are presented.

A PRIORI BOUNDS FOR GLOBAL SOLUTIONS OF HIGHER-ORDER SEMILINEAR PARABOLIC PROBLEMS

In this paper, we derive a priori bounds for global solutions of 2m-th order semilinear parabolic equations with superlinear and subcritical growth conditions. The proof is obtained by a bootstrap argument and maximal regularity estimates. If n≥ 10/3m, we also give another proof which does not use maximal regularity estimates.