Fast Interior Penalty Methods

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This project will develop fast numerical methods for fourth and higher order partial differential equations using the interior penalty approach. The interior penalty approach has advantages over the classical approaches that use conforming, nonconforming or mixed finite elements in terms of the computational complexity, the existence of natuaral hierarchies of elements, the preservation of the symetric positive definiteness of the continuous problem, and the ease of deriving convergent schemes for complicated problems. Another significant advantage of interior penalty methods for higher order problems is due to the fact that discontinuous finite elements for higher order problems are also suitable for lower order problems. Therefore multigrid algorithms for interior penalty methods can be developed recursively through the hierarchy of elliptic problems. Namely, multigrid algorithms for second order problems can be embedded naturally in multigrid algorithms for fourth order problems, which can then be embedded naturally in multigrid algorithms for sixth order problems, and so on. The performance of these multigrid methods for higher order problems is comparable to the performance of multigrid methods for second order problems. This project will initiate a comprehensive study of interior penalty methods for higher order problems together with multigrid, domain decomposition and adaptive algorithms that will provide fast solvers for the resulting discrete problems. The results of this project will make it feasible to solve problems of order six and higher on general domains. Applications of these methods to strain gradient elasticity, plate buckling, the Monge-Ampere equations and the Cahn-Hilliard equations will also be investigated.The fast algorithms developed in this project will make it practical for scientists and engineers to model complex phenomena by higher order partial differential equations. These algorithms will enhance the performance of numerical simulations in diverse areas such as structural mechanics, fluid mechanics, image processing, nanoscience, geometric optics, meteorology, optimal transport, differential geometry, and crystal growth, among many others.