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来描述，而是用更复杂的关系来描述，称为不等式(说明一个量大于或小于另一个量)。用数值方法求解一个不等系统可能是非常具有挑战性的。这一研究项目旨在为一类被称为半变分不等式的数值解发展严格而全面的数学理论以及可靠而有效的数值方法。除了计算数学的进步外，这项研究还将通过应用于井下抽水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根据测量的地面条件来推断井下抽油泵的条件。对于斜井(非垂直)油井，由于斜井中复杂的摩阻和动力学，目前还没有开发出可靠的模拟方法。合适的模型可以用半变分不等式的形式来表示，这一研究项目有望为斜井抽油条件的诊断提供实用的数值模拟。与石油行业研究人员的合作将确保将新的数学结果转化为应用。预计该项目的成果将有助于正确控制井下油泵，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，因此将接受高级数学和数值方法方面的培训，以解决应用中具有挑战性的重要问题。力学中的不等式问题可以分为两大类：一类是涉及凸能量泛函(势)的变分不等式问题，另一类是涉及非光滑和非凸能泛函(超势)的半变分不等式问题。通过半变分不等式的表述，可以成功地处理涉及非单调、非光滑和多值本构关系、力和边界条件的问题。在过去的三十年里，半变分不等式被证明在许多学科中都非常有用，从非光滑的力学、物理学、工程学到经济学。然而，在半变分不等式的数值分析方面所做的工作相对较少。在这个研究项目中，将发展一个全面的理论来数值求解各种半变分不等式，包括几类椭圆型、抛物型和双曲型半变分不等式。对于每一族半变分不等式，将引入基于有限元方法的空间离散和基于有限差分的时间离散的数值格式。在基本解的正则性下，给出了数值解的收敛性，并在适当的解的正则性假设下，给出了误差估计。当使用线性元素时，误差估计将是最优阶的。数值实验将用来说明理论预测的收敛阶。该项目的结果(如后验误差分析、自适应算法和间断Galerkin方法)将为进一步发展数值方法来求解半变分不等式奠定坚实的基础。

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.