파티션 함수의 크기가 무요소해법에 미치는 효과에 대한 고찰

Abstract

There is a tight link between the size of flat-top region and the gradient of the partition of unity function. Recent studies show that the condition number of the system can grow arbitrarily large with linear finite element shape functions as a partition of unity even though local approximation functions are linearly independent. As a result, a flat-top partition of unity function is introduced to avoid linear independence. The wide flat-top region in the partition of unity function has been known to ensure the linear independence of mesh-free basis functions. The wider the flat-top is, the smaller so better in the condition number. However, the wider flat-top partition of unity function could result in a large gradient of the partition of unity function as well as large condition number while having a strongly linear independent local basis functions. We demonstrate this phenomenon in a simple Poisson model problem when the flat-top region expands and starts to fill up the support of the partition of unity functions. For the particular problems we have considered, we find that there is an optimal size for the flat-top region in the support of partition of unity function.