In numerical mathematics, relaxation methods are iterative methods for solving systems of equations, including nonlinear systems.
[1] Relaxation methods were developed for solving large sparse linear systems, which arose as finite-difference discretizations of differential equations.
[2][3][4] Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation, it resembles repeated application of a local smoothing filter to the solution vector.
Nonetheless, the study of relaxation methods remains a core part of linear algebra, because the transformations of relaxation theory provide excellent preconditioners for new methods.
One can first compute an approximation on a coarser grid – usually the double spacing 2h – and use that solution with interpolated values for the other grid points as the initial assignment.