Model adaptivity for porous media

误差估计理论在计算科学中具有极其重要的意义。局部时间和空间网格自适应有助于以较低的计算成本最小化离散化误差。在这个项目中，研究者和研究生们进行了模型自适应性的研究，用另一个更容易求解的偏微分方程耦合系统代替原来的模型。模型自适应的主要思想是对总误差进行适当的分割，从而对离散化和建模误差进行后验估计，并对局部和全局进行控制。这些思想的实现取决于特定的应用及其数值近似，以及相关的兴趣量，这些兴趣量指导网格和模型自适应。本项目将重点研究多孔介质，特别是多尺度和优先型多孔介质及其各种符合和不符合有限元公式的应用。研究人员将证明新的理论结果，并将开发新的自适应算法，其原型实现将在公共领域提供。该项目将为一种新的快速和精确的计算方法奠定基础，用于模拟地球表面下发生的自然现象，如受污染的水流过复杂的地质构造。虽然传统方法使用固定的模型作为计算模拟的基础，但本研究项目中提出的新思想将允许丢弃模型中那些计算成本高但对解决方案没有重大影响的元素。该项目中提出的新方法将从理论和实践的角度进行评估，并可用于取代昂贵的灵敏度和参数研究。该项目的成果也将适用于其他学科，如多尺度材料的工程设计，中子输运的大规模模拟和生物医学应用。本科生和研究生将参与研究，并将在新的和传统的自适应计算方法及其应用方面进行培训。

The theory of error estimation is of paramount importance to thecomputational sciences. Local temporal and spatial grid adaptivityhelp to minimize the discretization error at a low computational cost.In this project the investigator and graduate students pursue a studyof model adaptivity in which the original model represented by acoupled system of partial differential equations is replaced byanother one which is easier to solve. The main idea of modeladaptivity is to use an appropriate splitting of the total error sothat the discretization and the modeling error are estimateda-posteriori and controlled locally and globally. Realization of theseideas depends on a particular application and its numericalapproximation as well as on the relevant quantities of interest whichguide both the grid and model adaptivity. This project will focus onapplications in porous media, in particular on those of multiscale andpreferential type, and on their various conforming and non-conformingfinite element formulations. The researchers will prove newtheoretical results as well as will develop new adaptive algorithmswhose prototype implementations will be available in public domain.The project will lay foundations for a new class of fast and accuratecomputational methods for simulation of natural phenomena occuringunder the Earth's surface such as flow of contaminated water throughand over complex geological formations. While traditional methods usea fixed model of such phenomena as a basis for computationalsimulations, the novel ideas proposed in this research project willallow to discard those elements of a model which are computationallycostly but which do not have significant impact on the solutions. Thenew methods proposed in the project will be evaluated from boththeoretical as well as from practical point of view and can be used inplace of costly sensitivity and parameter studies. The results of theproject will be also applicable to other disciplines such asengineering design of multiscale materials, large scale simulations ofneutron transport and biomedical applications. Undergraduate andgraduate students will be involved in research and will be trained inthe new and traditional adaptive computational methods and theirapplications.