Mixed finite elements and smooth approximations for partial differential equations

Northern Illinois University, Dekalb IL

Investigators

Abstract

The project is directed towards the development, analysis and improvement of numerical methods for the elasticity equations and Monge-Ampere type equations. The research in methods for the linear elasticity equations will focus on improvement of mixed finite element methods. Very simple elements with weakly imposed symmetry have been recently developed on triangular and tetrahedral meshes but these elements have yet to be extended to quadrilateral, 2D and 3D rectangular and hexahedral meshes which are often favored by practitioners. This proposal will use the technique of constructing piecewise polynomial exact sequences for the development of stable mixed finite elements on the above mentioned meshes. The second area of study is the construction and analysis of smooth approximations to Monge-Ampere type equations and the application of the methods developed to the solution of problems from science and engineering involving Monge-Ampere type equations. Several approaches will be followed, including global optimization ones. As it is well known the Monge-Ampere type equations, like other fully nonlinear partial differential equations do not possess in general smooth solutions but several of the approximation schemes, e.g. the vanishing moment methodology, require to work in spaces of smooth functions. The focus will be on the implementation and improvement of the spline element method, developed by the investigator and others, which uses multivariate splines for the solution of higher order partial differential equations. It leads to flexible, robust, efficient and accurate approximations allowing easy implementation, the flexibility of using polynomials of different degrees on different elements and the simplicity of a posteriori error estimates since the method is conforming. Mathematical modeling of physical phenomena have become the standard tool for the investigation of numerous problems in science and engineering. But often the resulting equations do not have solutions that can be represented by simple mathematical formulas. Hence the development of numerical methods and their analysis is essential to this process. This project adresses two types of equations which appear in fundamental problems but the impact of the methods developed here goes well beyond the particular applications being considered. The elasticity equations appear in many industrial, biological and engineering applications. The Monge-Ampere type equations appear in various geometric and variational problems, e.g. the Monge-Kantorovich problem. They also appear in applied fields such as meteorology, fluid mechanics, nonlinear elasticity, material sciences and mathematical finance. The development of the new methods from this project have the potential to put more competitive tools in the hands of the nation's scientists and engineers. The educational component of the project is that it will introduce a new generation of students to computational mathematics involving practical problems. Therefore this also contributes to national security and helps maintain the global scientific leadership position of the nation.