Numerical cubature for fully symmetric regions and adaptive methods for PDEs

这项研究计划有两个主要目标。其中一个方向继续了过去五年来在单空间变量时变偏微分方程数值解方面所做的工作。微分方程出现在许多物理模型中，并且解通常呈现出具有峰值或急剧变化的窄层。在许多情况下，这些快速变化的峰值或区域随时间移动。对于能够捕捉这些特征的近似解方法，离散模型必须适应解。将开发具有可靠误差估计的近似解的软件。我们将研究使用高阶插值的方法oflines的背景下，为了获得廉价和可靠的误差估计。使用一个等分布算法，我们将确保误差全局满足用户定义的errortolerance.The第二个推力，这也是一个长期的研究计划的延续，是建设的方法的数值计算的积分在multidimensionalregions具有指定的对称性。 这样的区域的例子是n维立方体，球体，单纯形，八面体，球面，和整个n-空间与一个numericweight函数。正如在较早的文件有一个强大的连接formulasfor表面的领域和单纯形。它的目的是调查之间的联系公式的八面体和单纯形。一个长期的计划是建立软件接受输入的对称性的区域，和一个要求的多项式次数，然后计算一致的结构的公式，建立定义方程，并解决他们。

This research programme has two principal thrusts. One direction continues work carried out in thelast five years on the numerical solution of time dependent partial differential equations in onespace variable. Differential equations occur in many physical models, and often the solutionexhibits narrow layers with peaks or sharp variations. In many cases these peaks or areas of rapidchange move in time. For an approximate method of solution to be able to catch these features, thediscrete model must adapt to the solution. Software to approximate to the solutions with reliable errorestimates will be developed. We will investigate the use of high order interpolants in a method oflines context, in order to obtain inexpensive and reliable error estimates. Using anequi-distribution algorithm we will ensure that the error globally satisfies a user-defined errortolerance.The second thrust, which also is a continuation of a long standing research programme, isthe construction of methods for the numerical evaluation of integrals over multidimensionalregions having specified symmetries.  Examples of such regions are the n dimensional cube,sphere, simplex, octahedron, spherical surface, and the whole of n-space with a symmetricweight function. As shown in an earlier paper there is a strong connection between formulasfor the surface of the sphere and for the simplex. It is intended to investigate connectionsbetween formulas for the octahedron and the simplex. A long term plan is to set up softwareto accept as input the symmetries of the region, and a requested polynomial degree, and thento compute consistent structures for the formulas, set up the defining equations, and solvethem.