Algorithms Of Three Class Of Elliptic Operator Problems

The elliptic operator (Laplace operator) issue from both the theoreticaldevelopment and practical applications used to have a very important posi-tion. In this thesis three class of elliptic operator problem associated with thealgorithm are studied.Firstly, an algorithm about the elliptic equation with free boundary valueproblem is investigated. The changes of the reactant concentration should beto consider, when an isothermal reaction was happened, then through thisprocess, we can get a nonhomogeneous Laplace equation. In the next, whatwe should do is to seek its non-negative solution. It is difcult to directly findsuch a solution, because there exists a unknown boundary in this problem.Fortunately, this free boundary value problem can be turned into a comple-mentary problem. Then we use some properties of the M-matrix and thediagonal function to design a non-interior continuation method to solve thefree boundary value problem. In general, the traditional non-interior methodneed to use a line search step to find the step length, while in this paper weremove the line search such that each iteration use a full step length. This isa new method. Numerical experiment shows that this algorithm has a greatefect.Secondly, an algorithm of an elliptic equation in optimal control problemis studied. It is diferent from the existing literature. The objective functioncontains a L~1-norm in this optimal control problem, while the L~1-norm isnot diferentiable. However, this problem can be transformed into a systemof nonsmooth equations which can be solved through using some smoothingstrategy. We design a grafting algorithm based on the limit of a smoothingfunction. This algorithm is divided into two parts. The first part is a correctedfixed-point method. If this part is used alone, the global convergence can beobtained, but in order to increase speed of the convergence the sequence begin to put into the second part when the sequence is near the true solution. Thisalgorithm can not only ensure the global convergence but also the locallyquadratic convergence.Finally, an algorithm about an elliptic-hyperbolic equation optimalcontrol with mixed state-control constrains is considered. A Slant-functionmethod (or semismooth method) is used to solve this problem. However,it is difcult to directly apply some theory of semismooth method. Wereconstruct the optimal condition and use the Lagrange multiplier theory tofind the dual of the original problem. Through the dual problem, we canget an equivalent nonsmooth equation system. By using Slant-function, wedesign a semismooth algorithm. Moreover, we analysis the convergence ofthis algorithm and obtain locally fast convergence.