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

有许多实际的物理情况，可以通过偏微分方程建模，其中一些系数在问题中，例如那些描述材料属性，是不连续的界面。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、地下水流和水库模拟、环境补救研究、晶体生长、波传播、沉积现象以及核燃料棒的制备。这些问题导致臭名昭著的困难，经典的数值方法作为一个直接的结果，缺乏光滑性。与此同时，使用固定的笛卡尔网格有效地解决界面问题的数值方法已经引起了相当大的关注，因为它们提供了一些计算上的优势和一些方法已经被追求。不幸的是，规则形状的离散化通常是有问题的，因为接口？削减？通过单元，而不考虑离散化的规则几何形状，这对所得到的近似的准确性具有强烈的负面影响。因此，至关重要的是提供计算误差估计，其量化在离散化和建模误差的各种来源方面的感兴趣的计算量的准确性。本项目将为几种离散界面方法开发变分有限元框架，这些方法确定模型的离散化和建模组件，并使用变分框架为指定的感兴趣量获得准确的后验误差估计。两个和三个空间维度的固定和演化问题将被考虑。有效的自适应离散化方法的发展平稳和发展的问题将得到解决。在这个建议中开发的方法和分析工具将提供一个强大的工具，系统地处理的问题，其中接口具有复杂的几何形状和材料性能变化很大的规模小于整体规模的离散化。该项目将产生一个系统的方法来获得可计算的和准确的误差估计的数量的利益，并进一步提供有关的相对贡献的离散化和建模所产生的错误的详细信息。

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.