RUI: Adaptively Weighted Finite Element Methods for PDEs and Optimal Least-Squares Metrics

偏微分方程数值方法的整体有效性可能会受到在相对较小的域子集上缺乏光滑性的解的严重限制。问题可能具有由域的几何形状引起的奇点；对流主导的区域可能形成内层或边界层；不连续的材料系数会引起急剧的梯度；或者当算子系数为奇异或简并时，解可能在内部点爆炸。本课题提出了用加权范数和内积代替标准范数和内积的加权有限元方法的系统研究。在最小二乘有限元设置中，这些权重函数用于重新定义误差最小的度量，因此，数值解的相对精度可以在整个域内以最优方式平衡。对于某些问题，正确选择权值可以在类似的非加权情况不收敛的情况下恢复收敛，而在其他问题中，通过一组适当的权值可以提高收敛速度。该项目将为广泛的线性和非线性问题开发鲁棒自适应方法，其中权重从网格细化策略中的粗尺度问题中选择。偏微分方程的有效数值解在整个应用科学中具有重要意义。有限元方法是解决广泛问题的一种流行和灵活的方法，并且在可变形固体力学、流体流动、输运和电磁学等应用中，对稳健、准确和高效的有限元算法的发展有着很高的需求。该项目旨在开发一类新的自适应有限元方法，其动机是加权范数最小二乘法和自适应网格细化算法的成功。鲁棒自适应算法通过将计算资源集中在问题中最具挑战性的方面，提高整体精度并减少计算时间，从而实现更好的数值模拟。例如，在冰川流动模型中，靠近地表和地面的冰区域需要比大部分内部冰更高的分辨率，而精确的模型必须能够以最优的方式分配计算工作。该项目的成功结果将增强当前对如何设计自适应算法以从粗尺度近似优化进化的理解。

The overall effectiveness of numerical methods for partial differential equations may be severely limited by solutions that lack smoothness on a relatively small subset of the domain. Problems may have singularities induced by the geometry of the domain; convection dominated regimes may result in interior or boundary layers; discontinuous material coefficients can cause sharp gradients; or solutions may blow up at interior points when operator coefficients are singular or degenerate. This project proposes a systematic study of weighted finite element methods where standard norms and inner products are replaced with weighted norms and inner products. In the least-squares finite element setting, these weight functions serve to redefine the metric under which the error is minimized and, as such, the relative accuracy of the numerical solution can be balanced throughout the domain in an optimal way. For some problems, the right choice of weights can recover convergence where the analogous nonweighted case does not converge and, in other problems, convergence rates are enhanced by an appropriate set of weights. This project will develop robust adaptive methods for a wide class of linear and nonlinear problems, where the weights are chosen from coarse scale problems within a mesh refinement strategy.The efficient numerical solution of partial differential equations is of great importance throughout the applied sciences. Finite element methods constitute a popular and flexible approach to solving a wide range of problems, and the development of robust, accurate, and efficient finite element algorithms is in high demand for applications including mechanics of deformable solids, fluid flow, transport, and electromagnetics. This project aims to develop a new class of adaptive finite element methods that are motivated by the success of weighted-norm least squares methods and adaptive mesh refinement algorithms. Robust adaptive algorithms allow for better numerical simulations by focusing computational resources on the most challenging aspects of the problem, increasing overall accuracy and decreasing computational time. In models of glacier flow, for example, regions of the ice near the surface and the ground require more resolution than the majority of the interior ice, and accurate models must be able to allocate the computational work in an optimal way. Successful results from this project will enhance the current understanding of how adaptive algorithms can be designed to evolve optimally from coarse scale approximations.