多空间与多层BDDC
Multispace and Multilevel BDDC
AI总结：
本文提出抽象的多空间BDDC方法族，并给出基于加性Schwarz预条件理论的条件数界，将多层BDDC作为特例，得到任意固定层数和二维/三维标量椭圆问题的多对数条件数界。
摘要：
BDDC方法是平衡族迭代子结构方法中用于求解椭圆边值问题离散化产生的大型线性代数方程组的最先进方法。在子结构数量众多的情况下，精确求解粗问题成为瓶颈。由于BDDC中的粗问题与原问题结构相同，递归应用BDDC方法仅近似求解粗问题是直接的。本文提出一个新的抽象多空间BDDC方法族，并从抽象加性Schwarz预条件理论给出条件数界。多层BDDC作为多空间BDDC的特例处理，并给出了抽象多层条件数界。抽象界对任意固定层数和二维、三维空间有限元离散的标量椭圆问题给出了多对数条件数界。数值实验验证了理论。
英文摘要：
BDDC method is the most advanced method from the Balancing family of iterative substructuring methods for the solution of large systems of linear algebraic equations arising from discretization of elliptic boundary value problems. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. Since the coarse problem in BDDC has the same structure as the original problem, it is straightforward to apply the BDDC method recursively to solve the coarse problem only approximately. In this paper, we formulate a new family of abstract Multispace BDDC methods and give condition number bounds from the abstract additive Schwarz preconditioning theory. The Multilevel BDDC is then treated as a special case of the Multispace BDDC and abstract multilevel condition number bounds are given. The abstract bounds yield polylogarithmic condition number bounds for an arbitrary fixed number of levels and scalar elliptic problems discretized by finite elements in two and three spatial dimensions. Numerical experiments confirm the theory.
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