使用多边形元素的稀疏算子适应小波分解用于多尺度有限元问题
Sparse Operator-Adapted Wavelet Decomposition Using Polygonal Elements for Multiscale FEM Problems
AI总结：
本文提出了一种基于多边形元素的稀疏算子适应小波分解方法，用于高效求解多尺度有限元问题。
摘要：
我们开发了一种基于稀疏多尺度算子适应小波分解的有限元方法（FEM），该方法在通过粗化过程获得的无结构多边形网格层次上进行。我们的方法解耦了不同的分辨率层次，允许每个尺度独立求解并添加到整个解中，而无需重新计算更粗的层次。在最细的层次上，网格由三角形元素组成，这些元素在每一步几何粗化以形成凸多边形元素。平滑的域区域通过较少、较大的多边形元素求解，而高梯度区域则由较小的元素表示，从而通过适应性提高内存效率。所提出的算法通过一系列层次稀疏线性代数操作计算解，具有近线性计算复杂度。
英文摘要：
We develop a sparse multiscale operator-adapted wavelet decomposition-based finite element method (FEM) on unstructured polygonal mesh hierarchies obtained via a coarsening procedure. Our approach decouples different resolution levels, allowing each scale to be solved independently and added to the entire solution without the need to recompute coarser levels. At the finest level, the meshes consist of triangular elements which are geometrically coarsened at each step to form convex polygonal elements. Smooth field regions of the domain are solved with fewer, larger, polygonal elements, whereas high-gradient regions are represented by smaller elements, thereby improving memory efficiency through adaptivity. The proposed algorithm computes solutions via sequences of hierarchical sparse linear-algebra operations with nearly linear computational complexity.
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