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

提高抛物线和椭圆系统的 Hp 自适应方法的性能

• 批准号: 0203154
• 负责人: Peter Moore
• 金额: $ 11.1万
• 依托单位: Southern Methodist University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0203154&HistoricalAwards=false
• 关键词: Enhancing; Performance; Hp; adaptive; Methods

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.

研究人员将开发工具来增强抛物线和椭圆系统在三个空间维度上的自适应解决方案。 一种新的后验误差估计策略，即基于插值误差的误差估计，在一维中开发，将扩展到三维。该策略不仅为当前网格，而且为元素比当前网格高一级的网格提供渐近精确的误差估计。 估计涉及从有限元解获得适当解导数的近似值，因此易于计算。 它们将增强 hp-refinement 代码的可靠性。 研究人员还将通过开发大型线性系统的新装配和求解算法来提高效率。 一种新的不完全汇编算法在非自适应设置中看起来很有前途，将扩展到 HP 自适应代码。 不完全组装类似于不完全因式分解，每行仅存储固定数量的非零条目，并且使用下降容差来丢弃小值。 将构建变量阶变量步骤隐式/显式求解器。 将研究一种根据过去的历史为瞬态问题的近似逆预处理器选择良好初始猜测的方法。这些策略将嵌入研究人员之前开发的 HP 细化代码中，并在一组反应扩散系统上进行测试。 许多重要的物理和生物过程可以通过椭圆和抛物型偏微分方程来建模，特别是反应扩散类型的方程。 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。 解决此类问题通常需要大量的计算资源，包括时间和存储，特别是当问题必须在三个空间维度上解决时。事实证明，自适应方法是有效的，它可以自动将这些资源分配到必须增强解决方案的领域，同时减少解决方案变化不大的领域的工作量。它们还通过将资源分配基于对误差的适当估计来提高解决方案的可靠性。 该提案的目的是通过两种方式显着改进自适应方法。首先，将开发一种新的误差估计策略，该策略更便宜并且提供更多信息作为资源分配的基础，从而提高可靠性。 第二个涉及创建几种新方法来求解大型方程组，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更精细的细节。