Parallel Multilevel Elliptic Preconditioners

近年来，人们对结合共轭梯度法求解椭圆型问题的多级预条件的研究越来越感兴趣。部分动机是这些方法除了通常具有良好的收敛性外，还具有高度并行性，这是大多数经典预条件（例如SSOR，不完全因数分解和多项式预条件）所缺乏的组合。作者之前在这一领域的工作使他们相信，这是一种非常有前途的方法，可以在大规模并行计算机上构建求解大型椭圆差分方程的高效鲁棒迭代方法。在这一领域的继续研究，将包括对大规模并行计算机（如CM.2）的实现问题的仔细研究，对不同多级预条件和一类域分解方法之间关系的理论研究，混合多级方法保留了不完全分解方法相对于非完全分解方法的良好性质。光滑系数，以及超越自伴随二阶椭圆算子的扩展（例如不定算子，对流扩散算子和双调和算子）。

In the last few years, there has been an increased interest in the study of multilevel preconditioners for solving elliptic problems in conjunction with the conjugate gradient method. Part of the motivation is that these methods are highly parallelizable, in addition to usually possessing good convergence properties, a combination that most of the classical preconditioners (e.g. SSOR, incomplete factorizations and polynomial preconditioners) lack. The authors' previous work in this area has convinced them that this is a very promising approach for constructing efficient and robust iterative methods for solving large elliptic difference equations on massively parallel computers. Continued research in this area, will include a careful study of implementation issues for massively parallel computers such as the CM.2, a theoretical study of the relationships among different multilevel preconditioners and the class of domain decomposition methods, hybrid multilevel methods which retain the nice properties of incomplete factorization methods with respect to non.smooth coefficients, and extensions beyond self-adjoint second order elliptic operators (e.g.indefinite, convection-diffusion and biharmonic operators.)