Collaborative Research: A posteriori error analysis and adaptivity for discontinuous interface problems

有许多实际的物理情况可以用偏微分方程来建模，其中问题中的一些系数，例如描述材料性质的系数，在界面上是不连续的。这些问题出现在非常广泛的科学和工程学科中，包括计算生物学、地下水流动和水库模拟、环境修复研究、晶体生长、波传播、沉积现象和核燃料棒的制备。由于缺乏平滑性，这类问题给经典数值方法造成了严重的困难。与此同时，利用固定笛卡尔网格有效解决界面问题的数值方法也引起了相当大的关注，因为它们提供了许多计算优势，并且已经追求了许多方法。不幸的是，规则形状的离散化通常是有问题的，因为界面切割？通过单元而不尊重规则几何的离散化，这对所得近似的准确性有强烈的负面影响。因此，根据离散化和建模误差的各种来源，提供计算误差估计来量化感兴趣的计算量的准确性是至关重要的。该项目将为几种离散接口方法开发变分有限元框架，这些方法可以识别模型的离散化和建模组件，并使用变分框架对指定的感兴趣数量得出准确的后验误差估计。将考虑二维和三维空间中的平稳和进化问题。将讨论针对平稳问题和进化问题的有效自适应离散化方法的发展。本提案中开发的方法和分析工具将为系统处理界面具有复杂几何形状和材料特性在小于离散化总体规模的范围内变化很大的问题提供强大的工具。该项目将产生一种系统的方法，为感兴趣的数量得出可计算和准确的误差估计，并进一步提供有关离散化和建模产生的误差的相关贡献的详细信息。

There are many practical physical situations that can be modeled by partial differential equations in which some of the coefficients in the problem, e.g. those describing material properties, are discontinuous across an interface. Such problems arise in a very broad range of scientific and engineering disciplines, including computational biology, ground water flow and reservoir simulation, environmental remediation studies, crystal growth, wave propagation, sedimentation phenomena, and the preparation of nuclear fuel rods. Such problems cause notorious difficulties for classic numerical methods as a direct result of the lack of smoothness. At the same time, numerical methods for efficiently solving interface problems using a fixed Cartesian grid have attracted considerable attention because they offer a number of computational advantages and a number of approaches have been pursued. Unfortunately, regular-shape discretizations are generally problematic because the interface ?cuts? through the cells without respecting the regular geometry of the discretization, which has a strongly negative impact on the accuracy of the resulting approximations. Consequently, it is critically important to provide computational error estimates that quantify the accuracy of a computed quantity of interest in terms of various sources of discretization and modeling error. This project will develop variational finite element frameworks for several discrete interface methods that identify both a discretization and a modeling component to the model and use the variational framework to derive accurate a posteriori error estimates for specified quantities of interest. Both stationary and evolutionary problems in two and three space dimensions will be considered. The development of efficient adaptive discretization methods for both stationary and evolution problems will be addressed. The methodology and analytic tools to be developed in this proposal will provide a powerful tool for the systematic treatment of problems in which interfaces have complex geometry and the material properties vary considerably on a scale smaller than the overall scale of the discretization. The project will yield a systematic approach to deriving computable and accurate error estimates for quantities of interest and further, provide detailed information about the relative contributions to the error arising from discretization and modeling.