Robust Multilevel Methods for General Symmetric Positive Definite Operators

An abstract robust multilevel method for solving symmetric positive definite systems resulting from discretizing elliptic partial differential equations is developed. The term “robust” refers to the convergence rate of the method being independent of discretization parameters, i.e., the problem size, and problem parameters. Important instances of such problem parameters are in particular (highly varying) coefficients. The method belongs to the class of (nonlinear) algebraic multilevel iterations (AMLI). The crucial ingredient for obtaining robustness is the construction of a nested sequence of spaces based on local generalized eigenvalue problems. The method is analyzed in a rather general setting and is applied to the scalar elliptic equation, the equations of linear elasticity, and equations arising in the solution of Maxwell’s equations. Numerical results for the scalar elliptic equation are presented showing its robust convergence behavior and large coarsening factors in the sequence of nested spaces.