Hemivariational Inequalities: Numerical Methods and Applications

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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.