The purpose of this paper is to delineate general algebraic coarsening techniques that can be employed for solving discretized partial differential equations by algebraic multigrid algorithms on unstructured grids or even on structured grids, when the coarse grid can no longer be structured or when the partial differential equation has highly disordered coefficients (for example Dirac equations in critical gauge field).
The description of coarsening schemes is given for a linear system of equations but the methods can be generalized to nonlinear and nondeterministic problems. The new concept introduced by the author is “localizability” which considers that each unknown of the linear system could be assigned a location in a low-dimensional space, such that each equation in the system involves only neighboring unknowns.
The paper re-examines various aspects of algebraic multigrid solvers and proposes new approaches for a relaxation algorithm, for interpolation, and for convergence acceleration by recombining iterants.