Mathematical Sciences: Development and Applications of Iterative Methods for Solving Nonlinear Elliptic Eigenvalue Problems

有了这个奖项，首席研究员将继续 发展基于迭代技术的数值方法， 一类非线性椭圆特征值的近似解 物理学和工程学中出现的问题。 什么设置这 这类问题，使它有趣的是， 要求近似的解必须满足 附加积分约束。 大多数数值方案， 当前使用椭圆方程的近似解， 满足域边界上的条件，而在此 授予主要研究者必须修改现有方案 或发明新的，以确保满意的， 积分约束 这样的特征值问题出现在 非线性波理论，有一个真正的需要， 可靠的数值方法来生成精确的近似 解决方案 在许多物理现象的数学模型中， 所考虑的过程被实现为 满足一定附加条件的微分方程 条件 例如，可以知道进程的状态 在某个初始时间或沿着域的边界。 在 在这些情况下，存在可靠的数值方法， 借助计算机得到精确的近似解。 在 这项补助金的主要研究者将研究的过程， 满足积分约束，而不是更熟悉的 初始条件或边界条件，因此她将 必须开发新的数值方法来找到解决方案。

With this award the principal investigator will continue to develop numerical methods based upon iterative techniques for the approximate solution of a class of nonlinear elliptic eigenvalue problems that arises in physics and engineering. What sets this class of problems apart and makes it interesting is the requirement that the solutions to be approximated must satisfy an additional integral constraint. Most numerical schemes in current use approximate solutions of elliptic equations that satisfy conditions on the boundary of the domain, while in this grant the principal investigator has to modify existing schemes or invent new ones in order to assure satisfaction of the integral constraint. Such eigenvalue problems arise in the theory of nonlinear waves and there is a genuine need for reliable numerical methods to generate accurate approximate solutions. In many mathematical models of physical phenomena the process under consideration is realized as the solution of a differential equation that satisfies certain additional conditions. For example, one may know the state of the process at some initial time or along the boundary of the domain. In these cases there exist reliable numerical methods that provide accurate approximate solutions with the aid of a computer. In this grant the principal investigator will study processes that satisfy an integral constraint, instead of the more familiar initial or boundary conditions alluded to above, and so she will have to develop new numerical methods to find the solution.