The Algorithm Analysis For Solving Obstacle And Degenerate Parabolic Problems

For many years, numerical method for differential equation, numerical approximationand numerical linear algebra have played roles of equal importance.However in recent years, as a result of rapid development in computer technology,the numerical method has demonstrated its ever increasing importance. In termsof mathematics, a number of important conclusion have shown the existence ofsolution of differential equation even under very general conditions. But in reality,what is need is not the existence of solution in mathematics, but the values orapproximation corresponding to specific solution of independent variable withina certain range of the definition-such a set of numerical values in a certain senseis called numerical solution of the differential equation. The process of lookingfor numerical solution is known as numerical solution for differential equation.Using numerical methods for differential equation problems, which has a historyof 150 years, came into being along with differential equation.Numerical solution methods are used to solve values of unknown functionsof each individual discrete points. First, we divide the whole range of definitionsinto a number of smaller regions in order to find approximation in points of eachsmall region or pieces. Then, for the sake of convenience, it is necessary to executesubdivision in field of definitions. Accordingly, original differential equation leadto the formation of recursion formulas or equations of function values for thesediscrete points or pieces, when their unknown variables are not a continuousfunction, but some kind of combination of a certain number of discrete unknownvalues. This process is called discretization of differential equation. If the discretesystem is of a recursive formula, it needs several initial values for calculation, orif it is a set of equations, then the number of its equations is often fewer than thatof unknown variable. In that case, more extra equations are needed for solvingthe solution through initial and boundary conditions of differenial equation.For the discrete system, what we primarily concern is that whether thesystem can be solved or not, that is, the existence and uniqueness of solution. Meanwhile we also concern the problems how large the error is when the diameterof region subdivision approaches zero and how fast the rate is when the errorapproaches zero, namely, the problems of convergence property and convergencespeed of solution. When the data from the outside world has noise, the resultsobtained may seriously deviate from the inherent solution of discrete systems, orstability of solution. This dissertation mainly investigates four kinds of differentialequations, each of which possesses its relevant model in practical application.Algorithms are presented for each differential equation and for each algorithm, aseries of problems mentioned above are discussed.This dissertation mainly studies four differential equations: obstacle problems,two phase Stefan type problems, nonlinear second order two point boundaryproblems and biharmonic equation.The second chapter is concerned about solving partial different equations ofthe obstacle problems by means of variational inequalities. First, we introducethe variational equalities, variational inequalities, variational inequality of thefirst kind, variational inequality of the second kind, complementarity problem,Newton iterative, the background and present situation of the obstacle problems,etc. Then we propose some iterative algorithms for the obstacle problemsdiscretized by the finite difference method. The proposed iterative algorithmshave the monotone property and uniform convergence, which converges into thesolution of the obstacle problems.Theorem 0.0.1. Let u_i,j⁽⁰⁾∈(?),u_i,j⁽⁰⁾=0, (i,j)∈(?)Ω_h. Then we have themonotone property of the solution of (2.37)And m→∞,u_ij^(m)→u_ij the solution of problem (2.44).In the third chapter we discuss Two Phase Stefan Type Problems, namelydegenerate parabolic initial-boundary value problem. First, we introduce Uzawaalgorithm, preconditioned Uzawa algorithm, inexact Uzawa algorithm, currentsituation of the degenerate parabolic initial-boundary value problem; Degenerateparabolic initial-boundary value problem can be equivalently rewritten as a discrete variational inequality of the second kind: Find u_h∈V_h such thatFor this variational inequality, we present a kind of standard Uzawa algorithm.Based on this algorith we propose the improved Uzawa algorithm. Convergenceproperty and convergence rate are analyzed for two algorithms respectively. Forthe improved Uzawa algorithm, we haveTheorem 0.0.2. Assume the family of the finite element triangulations is quasiuniform.Then there exists a constant c₀＞0 independent of the discretizationparametersÏ„and h such thatTherefore, ifÏ„h^-2 is kept bounded, then the optimal convergence rate of Algorithm3.4.5 is uniformly bounded and far away from 1.In the latter of this chapter, we also propose a kind of relaxation algorithm.Error estimates and convergence rate are also analyzed.In the last chapter, we try to study the nonlinear second order two pointboundary problems and biharmonic equations.First we introduces the background, classification, current situation of thenonlinear second order two point boundary problems and Pade approximationmethod, then proposes a symmetric compact finite difference method for solvingnonlinear second order two point boundary problems. This method can deducethe different schemes for the different parameter m . Numerical examples areillustrated for the method. For error estimates we can state as the followingtheorem .Theorem 0.0.3. Assume y(t) is the exact solution of equation (4.3), y_k is thesolution of Schemeâ… (4.11) and function f{t,y) satisfies the Lipchitz condition(4.14) with respect to y and L＜(?). Setε_k=y(t_k)-y_k, then we have theerror estimates of Schemeâ…  In the last part of this chapter, the background and current situation of thebiharmonic equations are introduced. Using the relation between finite differenceand finite element methods, we analyze the error estimates of a kind of two-dimensioncompact finite difference method. At last, a three-dimension compactfinite difference method is also suggested.