Dual Finite Element Methods for Challenging Computations

本研究计画发展一种新的偏微分方程之有限元素法。 有限元模型是科学计算中的强大工具，其应用包括生物医学工程、环境工程、材料和制造、结构工程等。 虽然目前的方法已成功地用于这些应用中的许多问题，但仍存在大量具有数值挑战性的问题，如多物理场和生物系统中的界面问题，以及超材料传输问题中的符号变化问题。 除了推进计算数学领域的知识外，该项目还将为当前方法难以计算准确和有效近似的问题提供新的有效工具。 这个项目的成功将导致新的计算方法与各种各样的应用。该项目还将通过培养研究生和本科生来支持教育。该项目的目的是开发一种新的有限元方法，用于精确有效地近似对偶变量。 为了精确逼近对偶变量，传统的数值方法，如最小二乘或混合有限元法，大大增加了自由度，由此产生的代数方程可能是不确定的。 提出的研究开发了一种新的方法，近似只有对偶变量，而不近似的主要变量。 这将导致一个较小的问题大小和由此产生的代数方程具有对称和正定矩阵。 将开发各种误差估计，包括自适应程序的后验误差估计。这种基于对偶的有限元法在求解奇异摄动问题时表现出上级性能。 该项目将解决这种方法在数值上具有挑战性的问题，如不连续系数，符号变化问题，线性弹性和Stokes方程的应用。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.