Hemivariational Inequalities: Numerical Methods and Applications

物理科学和工程中的某些系统不能用数学方程描述，而是用更复杂的关系描述，称为不等式（说明一个量大于或小于另一个量）。 用数值方法求解一组不等式可能非常具有挑战性。该研究项目旨在为一类称为半变分不等式的数值解开发严格而全面的数学理论，以及可靠而有效的数值方法。除了在计算数学方面取得的进展外，这项研究还将通过应用于井眼泵送系统的分析而对石油工业产生影响。石油工业使用数值模拟来基于测量的表面条件推断井下油泵条件。对于斜（非垂直）油井威尔斯，主要由于斜威尔斯井中复杂的摩擦和动态，尚未开发可靠的模拟方法。适当的模型可以铸造的半变分不等式的形式，该研究项目预计将促进实际有用的数值模拟诊断的油泵条件在斜井威尔斯。与石油工业研究人员的合作将确保将新的数学结果转移到应用程序中。该项目的研究结果有望帮助正确控制井下抽油泵，以节省能源，避免泵损坏。 研究生将积极参与研究的各个方面，因此将接受高层次数学和数值方法的培训，以解决应用中具有挑战性和重要性的问题。力学中的不等式问题可以分为两大类：一类是与凸能量泛函（势）有关的变分不等式，另一类是与非光滑非凸能量泛函（超势）有关的半变分不等式。通过半变分不等式的制定，涉及非单调，非光滑，多值本构关系，力和边界条件的问题可以成功地处理。在过去的三十年中，半变分不等式被证明是非常有用的各种学科，从非光滑力学，物理学，工程，经济学。然而，相对较少的工作，半变分不等式的数值分析已经完成。在这个研究项目中，将开发一个全面的理论，用于各种半变分不等式的数值解，包括几个家庭的椭圆，抛物和双曲半变分不等式。对于每个家庭的半变分不等式，数值方案将被引入基于有限元方法的空间离散和有限差分的时间离散。数值解的收敛性将在基本解的正则性下得到证明，并在适当的解的正则性假设下得到误差估计。当使用线性元时，误差估计将具有最优阶。将进行数值实验，以说明理论预测的收敛阶。从这个项目的结果（如后验误差分析，自适应算法和间断Galerkin方法）将形成一个坚实的基础，进一步发展数值方法来解决半变分不等式。

Certain systems in physical sciences and engineering cannot be described by mathematical equations but are instead described by more complicated relations known as inequalities (stating that one quantity is greater than or smaller than another). Solving a system of inequalities numerically can be very challenging. This research project aims to develop a rigorous and comprehensive mathematical theory, as well as reliable and efficient numerical methods, for the numerical solution of a class known as hemivariational inequalities. In addition to the resultant advances in computational mathematics, the research will have impact in the petroleum industry through application to analysis of borehole pumping systems. Numerical simulations are used by the petroleum industry to infer the downhole oil pump conditions based on measured surface conditions. For deviated (non-vertical) oil wells, methods for reliable simulation have yet to be developed, mainly due to the complicated friction and dynamics in deviated wells. Appropriate models can be cast in the form of hemivariational inequalities; this research project is expected to facilitate practically useful numerical simulations for diagnostics of oil pump conditions in deviated wells. Collaboration with researchers in the petroleum industry will ensure the transfer of the new mathematical results to applications. The results from the project are expected to help correctly control downhole oil pumps to save energy and avoid pump damage. Graduate students will actively participate in all aspects of the research and will thus be trained in high level mathematics and numerical methods on challenging and important problems from applications. Inequality problems in mechanics can be divided into two main classes: that of variational inequalities, which is concerned with convex energy functionals (potentials), and that of hemivariational inequalities, which is concerned with nonsmooth and nonconvex energy functionals (superpotentials). Through the formulation of hemivariational inequalities, problems involving nonmonotone, nonsmooth, and multivalued constitutive laws, forces, and boundary conditions can be treated successfully. During the last three decades, hemivariational inequalities were shown to be very useful across a wide variety of disciplines, ranging from nonsmooth mechanics, physics, and engineering, to economics. However, relatively little work on the numerical analysis of hemivariational inequalities has been done. In this research project, a comprehensive theory will be developed for the numerical solution of various hemivariational inequalities, including several families of elliptic, parabolic, and hyperbolic hemivariational inequalities. For each family of hemivariational inequalities, numerical schemes will be introduced based on the finite element method for spatial discretization and finite differences for temporal discretization. Convergence of the numerical solutions will be shown under the basic solution regularity, and error estimates will be derived under appropriate solution regularity assumptions. The error estimates will be of optimal order when linear elements are used. Numerical experiments will be performed to illustrate convergence orders predicted by the theory. Results from this project (such as a posteriori error analysis, adaptive algorithms, and discontinuous Galerkin methods) will form a solid foundation for further developing numerical methods to solve hemivariational inequalities.