Adaptive Lavrentiev Regularization of Singular and Ill-Conditioned Discrete Boundary Value Problems in the Robust Multigrid Technique

The paper presents a multigrid algorithm with the effective procedure for finding problem-dependent components of smoothers. The discrete Neumann-type boundary value problem is taken as a model problem. To overcome the difficulties caused by the singularity of the coefficient matrix of the resulting system of linear equations, the discrete Neumann-type boundary value problem is solved by direct Gauss elimination on the coarsest level. At finer grid levels, Lavrentiev (shift) regularization is used to construct the approximate solutions of singular or ill-conditioned problems. The regularization parameter for the unperturbed systems can be defined using the proximity of solutions obtained at the coarser grid levels. The paper presents the multigrid algorithm, an analysis of convergence and perturbation errors, a procedure for the definition of the starting guess for the Neumann boundary value problem satisfying the compatibility condition, and an extrapolation of solutions of regularized linear systems. This robust algorithm with the least number of problem-dependent components will be useful in solving the industrial problems.