A unified approach to the design and analysis of AMG
Provides a unified theoretical foundation for AMG methods, benefiting researchers and practitioners in numerical linear algebra and scientific computing.
This work presents a general framework for designing and analyzing two-level algebraic multigrid (AMG) methods, achieving uniform convergence for finite element discretizations of jump coefficient problems on shape-regular meshes.
In this work, we present a general framework for the design and analysis of two-level AMG methods. The approach is to find a basis for locally optimal or quasi-optimal coarse space, such as the space of constant vectors for standard discretizations of scalar elliptic partial differential equations. The locally defined basis elements are glued together using carefully designed linear extension maps to form a global coarse space. Such coarse spaces, constructed locally, satisfy global approximation property and by estimating the local Poincar{\' e} constants, we obtain sharp bounds on the convergence rate of the resulting two-level methods. To illustrate the use of the theoretical framework in practice, we prove the uniform convergence of the classical two level AMG method for finite element discretization of a jump coefficient problem on a shape regular mesh.