the major problem to solve such nearly singular systems boils down to treat the nearly singular operator given by robustly with respect to the positive, but small parameter . Here is symmetric semidefinite operator with large null space, while is a symmetric positive definite operator. There were many works to attempt to design a robust and fast multigrid method for such nearly singular problems. A general guide has been provided as a design principle to achieve parameters (e.g., mesh size and physical parameters such as Poisson's ratio that appear in the nearly singular operator) independent convergence rate of the multigrid method applied to such nearly singular systems, i.e., in each grid, a space decomposition based on which the smoothing is applied, has to be constructed so that the null space of the singular part of the nearly singular operator has to be included in the sum of the local null spaces, the intersection of the null space and the local spaces resulting from the space decompositions. Notes References G. P. Astrachancev (1971), An iterative method of solving elliptic net problems. USSR Comp. Math. Math. Phys. 11, 171–182. N. S. Bakhvalov (1966), On the convergence of a relaxation method with natural constraints on the elliptic operator. USSR Comp. Math. Math. Phys. 6, 101–13. Achi Brandt (April 1977), "Multi-Level Adaptive Solutions to Boundary-Value Problems", Mathematics of Computation, 31: 333–90. William L. Briggs, Van Emden Henson, and Steve F. McCormick (2000), A Multigrid Tutorial (2nd ed.), Philadelphia: Society for Industrial and Applied Mathematics, . R. P. Fedorenko (1961), A relaxation method for solving elliptic difference equations. USSR Comput. Math. Math. Phys. 1, p. 1092. R. P. Fedorenko (1964), The speed of convergence of one iterative process. USSR Comput. Math. Math. Phys. 4, p. 227. External links Repository for multigrid, multilevel, multiscale, aggregation, defect correction, and domain decomposition methods Multigrid tutorial Algebraic multigrid tutorial Links to AMG presentations Numerical analysis Partial differential equations Wavelets