Dual Finite Element Methods for Challenging Computations

Oklahoma State University, Stillwater OK

Investigators

Abstract

This research project develops a new finite element method(FEM) for the approximation solutions of partial differential equations. FEMs are powerful tools in scientific computing and their applications include biomedical engineering, environmental engineering, material and manufacturing, structural engineering, and more. While current methods are successfully used for many problems in these applications, there are plenty of numerically challenging problems such as interface problems arising from multiphysics and biological systems, and sign changing problems from the transmission problems with metamaterials. In addition to advancing knowledge within the field of computational mathematics, this project will provide new efficient tools for the problems which current methods have difficulties calculating accurate and efficient approximations. The success of this project will lead to new computational methods with a wide variety of applications. This project will also support education by training graduate and undergraduate students. The purpose of this project is to develop a new finite element method for accurate and efficient approximations of dual variables. For accurate approximations of the dual variables, traditional numerical methods such as least-squares or mixed FEMs increase the degrees of freedom significantly, and the resulting algebraic equations could be indefinite. The proposed research develops a new method that approximates only the dual variables without approximating the primary variable. This results in a smaller problem size and the resulting algebraic equations have symmetric and positive definite matrices. Various error estimates will be developed, including a posteriori error estimates for adaptive procedures. This dual based FEM shows superior performance when it is applied to singularly perturbed problems. This project will address the application of this approach to numerically challenging problems such as discontinuous coefficient, sign changing problems, linear elasticity, and Stokes equations. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.