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

研究者将开发工具，将增强自适应解决抛物线和椭圆系统在三维空间。一种新的后验误差估计策略，即基于插值误差的误差估计，将从一维扩展到三维。该策略不仅为当前网格提供了渐近精确的误差估计，而且为元素比当前网格高一度的网格提供了误差估计。估计包括从有限元解中获得适当解导数的近似值，因此易于计算。它们将提高hp-细化代码的可靠性。研究者还将通过开发大型线性系统的新装配和解决算法来提高效率。一种新的不完全汇编算法在非自适应环境下看起来很有前途，并将其扩展到hp自适应代码中。不完全汇编类似于不完全分解，因为每行只存储固定数量的非零项，并且使用丢弃容差来丢弃小值。建立变阶-变阶隐式/显式求解器。研究了一种基于过去历史的时间相关问题的近似逆预条件选择良好初始猜测的方法。这些策略将嵌入到先前由研究者开发的hp优化代码中，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用椭圆型和抛物型偏微分方程来模拟，特别是那些反应扩散型偏微分方程。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维空间中解决时。适应性方法已被证明是有效的，它自动地将这些资源定位到必须增强解决方案的区域，同时减少解决方案变化不大的区域的工作量。它们还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。本建议的目的是在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，该策略成本更低，并提供更多的信息，以作为资源分配的基础，从而提高可靠性。第二项涉及到创造一些新的方法来解决大型方程组，从而提高时间和存储效率。反过来，这意味着科学家和工程师将能够探索更现实的模型，而不是在数值分析的细节上深思熟虑。

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.