Enhancing the Performance of Hp-adaptive Methods for Parabolic and Elliptic Systems

研究人员将开发工具，以加强三维空间中抛物和椭圆系统的自适应解决方案。 本文将一种新的一维后验误差估计方法--基于插值误差的误差估计方法推广到三维。该策略不仅为当前网格，而且为元素比当前网格高一度的网格提供了误差的渐近精确估计。 估计涉及获得近似的适当的解决方案衍生物的有限元解决方案，因此，易于计算。 它们将提高hp-精化码的可靠性。 研究人员还将通过为大型线性系统开发新的组装和解决方案算法来提高效率。 一个新的不完全汇编算法，似乎有希望在非自适应设置将扩展到一个惠普自适应代码。 不完全汇编类似于不完全因式分解，因为每行只存储固定数量的非零条目，并且使用丢弃容差来丢弃较小的值。 将构建变阶变步长隐式/显式求解器。 将研究一种基于过去历史为时间相关问题的近似逆预条件子选择好的初始猜测的方法。这些策略将被嵌入到以前开发的研究人员和测试的一组反应扩散系统的hp细化代码。 许多重要的物理和生物过程可以用椭圆和抛物型偏微分方程来模拟，特别是反应扩散型方程。 一些例子包括化学和生物系统中的模式形成，心脏中的燃烧和电压传播。 解决这些问题通常涉及大量的计算资源，包括时间和存储，特别是当问题必须在三维空间中解决时。自适应方法已被证明是有效的，它自动将这些资源定位到解决方案必须增强的领域，同时减少解决方案变化不大的领域的工作。它们还通过基于对误差的适当估计来分配资源，从而提高了解决方案的可靠性。 该提案的目的是从两个方面显著改进自适应方法。首先，将开发一种新的误差估计策略，该策略更便宜，并提供更多的资源分配信息，从而提高可靠性。 第二个涉及到创建几个新的方法来解决大型方程组，从而提高时间和存储效率。 这反过来意味着科学家和工程师将能够探索更现实的模型，而不是仔细考虑数值分析的细节。

The investigator will develop tools that will enhance the adaptive solution of parabolic and elliptic systems in three space dimensions. A new a posteriori error estimation strategy, interpolation-error based error estimation, developed in one dimension will be extended to three dimensions. This strategy provides asymptotically exact estimates of the error not only for the current grid, but for a grid whose elements are one degree higher than the current one. The estimates involve obtaining approximations of appropriate solution derivatives from the finite element solution and are thus, easy to compute. They will enhance the reliability of hp-refinement codes. The investigator will also improve efficiency through the development of new assembly and solution algorithms for large linear systems. A new incomplete assembly algorithm that appears promising in a nonadaptive setting will be extended to an hp-adaptive code. Incomplete assembly is analogous to incomplete factorization in that only a fixed number of nonzero entries are stored per row and a drop tolerance is used to discard small values. A variable order-variable step implicit/explicit solver will be built. A method for selecting a good initial guess for an approximate inverse preconditioner for time-dependent problems based on past history will be investigated. These strategies will be embedded in an hp-refinement code developed previously by the investigator and tested on a set of reaction-diffusion systems. Many important physical and biological processes can be modeled by elliptic and parabolic partial differential equations, especially those of the reaction-diffusion type. Some examples include pattern formation in chemical and biological systems, combustion and voltage propagation in the heart. Solving such problems often involves significant computational resources, both time and storage, especially when the problems must be solved in three space dimensions. Adaptive methods have proven effective by automatically targeting these resources to areas where the solution must be enhanced while reducing the effort in areas where the solution is changing little. They also increase the reliability of the solution by basing resource allocation on appropriate estimation of the error. The aim of this proposal is to significantly improve adaptive methods in two ways. First a new error estimation strategy will be developed that is cheaper and provides more information on which to base resource allocation, thereby increasing reliability. The second involves the creation of several new methods to solve large systems of equations thus improving both time and storage efficiency. This in turn, means that scientists and engineers will be able to explore more realistic models rather than deliberate over the finer details of numerical analysis.