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.
New classes of numerical schemes for partial differential equations and related optimization problems have resulted from this project. One class of schemes can be applied to higher order problems such as the bending of thin plates in mechanics and the phase separation phenomenon in material science. A second class of schemes can be applied to fully nonlinear partial differential equations arising from differential geometry and optimal transport. Another class of schemes can be applied to two dimensional problems in electromagnetics. New fast solution methods involving multilevel techniques and parallel computing have also been developed for numerical schemes that can handle complicated meshes and multi-physics couplings, and for numerical schemes that can handle saddle point problems in structural and fluid mechanics. Several breakthroughs have been achieved in this project: (1) A general framework for the numerical analysis of fourth order variational inequalities has been developed, which fills a void in the literature that has existed for more than three decades. (2) A new approach to the numerical solution of two dimensional electromagnetic problems has been developed that would allow many existing methods for scalar equations to be used in computational electromagnetics. (3) A new class of multigrid methods for saddle point problems appearing in mechanics has been developed. These methods converge uniformly in the energy norm that underlies the physics and they also converge for general domains. (4) A new general approach to the numerical solution of fully nonlinear partial differential equations has been developed. The outcomes of this project will provide new tools for numerical simulations in science and engineering.
