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

研究人员科克伯恩和他的同事伯纳多·科克伯恩在一个合作项目中，为具有移动界面和长时间动力学的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。有限元方法长期以来一直被用来求解偏微分方程组。近年来，使用间断逼近空间的方法变得流行起来，特别是对于非定常对流扩散问题。两种这样的方法是由研究人员及其合作者开发的局部不连续Galerkin方法(LDG)和Godunov混合方法(GMM)。这些方法的优点是它们基于局部守恒和近似激波以及无伪振荡的尖锐梯度，这在许多对流扩散应用中是很重要的。它们也适合于并行计算。这两种方法都已在计算机上实现，并得到了先验误差估计；然而，还没有基于这些方法的自适应策略被开发出来。人们早就认识到，为了获得准确的解，在模拟过程中调整有限元网格和时间步长是可取的。为了成功地调整网格以保证实际误差低于给定的容差，必须开发测量作为网格、时间步长和计算解的函数的实际误差的后验误差估计。虽然对定常问题和协调有限元空间的这种估计有相当多的文献，但对非定常问题和不连续方法的研究很少。在这个项目中，研究人员和他们的同事开发了对流扩散方程的LDG和GMM方法的后验估计，重点放在三个重要的应用上：浅水流动，多孔介质和地表水中的化学反应传输，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方法；然而，也研究了其他可能更有效的特别方法。这些估计在应用和数值方法方面都是新颖的。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)的水流模式建模对于了解石油泄漏对环境的影响和疏浚对经济的影响很重要，也有助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具有重要意义。脑肿瘤的模型可以用于预测肿瘤的生长和检查潜在的治疗方法。这些应用虽然多种多样，但具有共同的数学特征，并可以利用类似的数值模拟方法。这些应用和方法缺乏可靠的、基于数学的工具，用于控制和调整模拟以满足用户感兴趣的特定精度标准；即可用于确定数值模拟是否确实反映物理现实的标准。该项目的目标是制定这样的标准，使模拟有效、准确和物理现实，并在基本数学、计算科学和应用的多学科方面培训未来的研究人员。

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.