Higher Order Variational Inequalities: Novel Finite Element Methods and Fast Solvers

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Variational inequalities appear in areas that involve differential equations and optimization. They are fundamental tools for the modeling of phenomena in science, engineering and finance that involve inequality constraints. The goal of this project is to design, analyze and implement reliable and efficient numerical algorithms for variational inequalities that involve higher order differential equations, with applications to optimal control and materials science.Novel finite element methods for higher order elliptic and parabolic variational inequalities will be developed together with fast solution techniques (adaptive, parallel and multilevel) for the resulting discrete problems. The emphasis is for problems on nonsmooth and nonconvex domains where the regularity of the solutions of the variational inequalities is much more subtle, and for problems on three dimensional domains where numerical computations are much more demanding. An important application is to optimal control problems constrained by elliptic partial differential equations. By reformulating these optimal control problems as fourth order variational inequalities for the state variable, various finite element methodologies(classical conforming and nonconforming finite element methods, discontinuous Galerkin methods, partition of unity methods, mixed finite element methods, etc.) and techniques (error estimators, local mesh refinements, inclusion of singularities in local approximation spaces, etc.) can be employed in their numerical solutions. The new numerical methods designed from this approach will be fundamentally different from the ones obtained by the traditional approach where the emphasis is on the control variable.