A Posteriori Error Estimates for Discontinuous Finite Element Methods Applied to Problems in Geosciences and Medicine

Cockburn 研究人员和他的同事Bernardo Cockburn在一个合作项目中，为移动界面和长时间动态问题开发了自适应数值方法。 这些问题涉及多个时间和空间尺度。 有限元法用于求解偏微分方程由来已久。 最近，使用不连续逼近空间的方法已经变得流行，特别是对于非定常对流扩散问题。 两个这样的方法是局部间断伽辽金方法（LDG）和Godunov混合方法（GMM），由研究人员和他们的合作者开发的。 这些方法的优点是它们基于局部守恒和近似激波以及没有虚假振荡的尖锐梯度，这在许多对流扩散应用中是重要的。 它们也适用于并行计算。 这两种方法都已实现计算，并推导出先验误差估计，但是，没有自适应策略的基础上，这些方法已经开发出来。 长期以来，人们已经认识到，在仿真期间调整有限元网格和时间步长对于获得准确的解是期望的。 为了成功地调整网格以保证实际误差低于给定的公差，必须开发后验误差估计，该后验误差估计测量作为网格、时间步长和计算解的函数的实际误差。 虽然大量的文献存在这样的估计稳定的问题和协调有限元空间，很少有研究已经做了非稳定问题和不连续的方法。在这个项目中，研究人员和他们的同事为对流扩散方程的LDG和GMM方法开发了后验估计，重点是三个重要的应用：浅水流动，多孔介质和地表水中的化学反应运输，以及脑肿瘤细胞生长和治疗的建模。 这些估计的基础是所谓的近似伴随方程的方法，然而，其他特设的方法，可能是更有效的也进行了研究。 这些估计是新颖的应用程序以及数值方法。 感兴趣的应用对工业、政府实验室和部门以及国家机构都很重要。浅水系统（如海湾和河口）的流动模式建模对于了解石油泄漏的环境影响和疏浚的经济影响等问题非常重要，还可用于跟踪飓风和其他极端天气事件期间的风暴潮。 模拟地下水和地表水中化学物质的迁移对于理解废物处理和污染修复非常重要。 脑肿瘤的建模可以用于预测肿瘤生长和检查潜在的治疗方法。这些应用虽然各不相同，但具有共同的数学特征，并且可以利用类似的数值模拟方法。 缺乏这些应用程序和方法是健全的，数学为基础的工具，用于控制和适应模拟，以满足特定的准确性标准的兴趣，用户，也就是说，标准，可用于确定是否数值模拟实际上反映了物理现实。 该项目的目标是制定这样的标准，使模拟高效，准确和物理现实，并培养未来的研究人员在基础数学，计算科学和多学科方面的应用。

Cockburn The investigator and his colleague Bernardo Cockburn, in a collaborative project, develop adaptive numerical methods for problems with moving interfaces and long-time dynamics. These problems incorporate multiple temporal and spatial scales. Finite element methods have long been used for solving partial differential equations. Recently, methods using discontinuous approximating spaces have become popular, especially for nonsteady convection-diffusion problems. Two such methods are the local discontinuous Galerkin method (LDG) and the Godunov-mixed method (GMM), developed by the investigators and their collaborators. These methods have the advantage that they are based on local conservation and approximate shocks and sharp gradients with no spurious oscillations, which is important in many convection-diffusion applications. They also lend themselves to parallel computation. Both methods have been implemented computationally, and a priori error estimates have been derived; however, no adaptive strategies based on these methods have been developed. It has long been recognized that adapting the finite element mesh and time-step during a simulation is desirable for obtaining an accurate solution. In order to successfully adapt the mesh to guarantee that the actual error is below a given tolerance, it is essential to develop a posteriori error estimates that measure the actual error as a funtion of the mesh, time step and the computed solution. While a substantial literature exists for such estimates for steady problems and conforming finite element spaces, little research has been done for nonsteady problems and discontinuous methods. In this project, the investigators and their colleagues develop a posteriori estimates for the LDG and GMM methods for convection-diffusion equations, with emphasis on three important applications: shallow water flow, chemically reactive transport in porous media and surface water, and the modeling of brain tumor cell growth and treatment. The basis for these estimates is the so-called approximate adjoint equation methodology; however, other ad-hoc methods which may potentially be more efficient are also investigated. These estimates are novel for the applications as well as the numerical methods. The applications of interest are important to industry, government laboratories and departments, and state agencies. Modeling of flow patterns in shallow water systems (e.g. bays and estuaries) is important for understanding, for instance, the environmental impacts of oil spills and the economic impacts of dredging, and can also be useful in tracking storm surges during hurricanes and other extreme weather events. Modeling of transport of chemical species in groundwater and surface water is important for understanding waste disposal and pollution remediation. The modeling of brain tumors can be useful in predicting tumor growth and examining potential treatments. These applications, though varied, share common mathematical characteristics and can utilize similar numerical simulation methodologies. Lacking for these applications and methodologies are sound, mathematically based tools for controlling and adapting the simulations to meet specific accuracy criteria of interest to the user; that is, criteria which can be used to determine whether a numerical simulation actually reflects physical reality. The goals of this project are to develop such criteria for making the simulations efficient, accurate and physically realistic, and to train future researchers in the underlying mathematics, computational science and multidisciplinary aspects of the applications.