Dual Finite Element Methods for Challenging Computations

本研究发展了一种新的偏微分方程解的有限元方法。FEMS是科学计算的强大工具，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。虽然目前的方法已经成功地应用于这些应用中的许多问题，但仍然存在大量具有数值挑战性的问题，如来自多物理和生物系统的界面问题，以及由超材料传输问题引起的符号变化问题。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的近似的问题提供新的有效工具。该项目的成功将导致新的计算方法的广泛应用。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，用于准确和有效地近似对偶变量。对于对偶变量的精确逼近，传统的数值方法，如最小二乘法或混合有限元方法，大大增加了自由度，所得到的代数方程可能是不定的。这项研究提出了一种新的方法，只近似对偶变量，而不近似主变量。这导致了较小的问题规模，并且所得到的代数方程具有对称和正定矩阵。将开发各种误差估计，包括用于自适应程序的后验误差估计。当应用于奇异摄动问题时，这种基于对偶的有限元方法表现出了优越的性能。这个项目将解决这种方法在数字挑战问题上的应用，例如不连续系数、符号变化问题、线性弹性和斯托克斯方程。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.