RUI: Adaptively Weighted Finite Element Methods for PDEs and Optimal Least-Squares Metrics
RUI:偏微分方程和最优最小二乘度量的自适应加权有限元方法
基本信息
- 批准号:1216297
- 负责人:
- 金额:$ 11.29万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2012
- 资助国家:美国
- 起止时间:2012-09-15 至 2015-08-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
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.
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。问题的奇点可能由区域的几何形状引起;对流主导的区域可能会导致内部或边界层;不连续的材料系数可能会导致急剧的梯度;或者当算子系数奇异或退化时,解可能在内部点处爆炸。本项目对加权有限元方法进行了系统的研究,将标准范数和内积替换为加权范数和内积。在最小二乘有限元设置中,这些权函数用于重新定义误差最小化的度量,因此,数值解的相对精度可以在整个域中以最佳方式平衡。对于某些问题,正确的权重选择可以在类似的非加权情形不收敛的情况下恢复收敛,而在其他问题中,适当的权重集合可以提高收敛速度。这个项目将开发一种稳健的自适应方法来解决广泛的线性和非线性问题,其中权重是在网格加密策略下从粗尺度问题中选择的。有效的偏微分方程组的数值解在整个应用科学中具有重要的意义。有限元方法是解决各种问题的一种流行而灵活的方法,对于可变形固体力学、流体流动、传输和电磁学等应用领域,开发健壮、准确和高效的有限元算法具有很高的要求。该项目旨在开发一类新的自适应有限元方法,其动力来自于加权范数最小二乘法和自适应网格加密算法的成功。稳健的自适应算法通过将计算资源集中在问题的最具挑战性的方面、提高总体精度和减少计算时间来实现更好的数值模拟。例如,在冰川流动模型中,地表和地面附近的冰区域需要比大多数内部冰更高的分辨率,准确的模型必须能够以最佳方式分配计算工作。该项目的成功结果将加强目前对如何设计自适应算法以从粗略规模近似进化到最优的理解。
项目成果
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