Nearly singular problems arise in a number of important physical and engineering applications. Simple, but important example of nearly singular problems can be found at the displacement formulation of [[linear elasticity]] for nearly incompressible materials. Typically, the major problem to solve such nearly singular systems boils down to treat the nearly singular operator given by <math>A + \epsilonvarepsilon M</math> robustly with respect to the positive, but small parameter <math>\epsilonvarepsilon</math>. Here <math>A</math> is symmetric [[Positive semidefinite matrix|semidefinite]] operator with large [[Kernel (linear algebra)|null space]], while <math>M</math> is a symmetric [[Definiteness of a matrix|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,<ref>Young-Ju Lee, Jinbiao Wu, Jinchao Xu and Ludmil Zikatanov, Robust Subspace Correction Methods for Nearly Singular Systems, Mathematical Models and Methods in Applied Sciences, Vol. 17, No 11, pp. 1937-1963 (2007)</ref> 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.
