If the dependent variables in a model have widely different magnitudes, the solver might have problems with the resulting ill-conditioned matrix. The scaling of the dependent variables also influences the weighted errors computed by the solvers. For instance, in a structural mechanics problem the displacements can be of the order of 0.0001 m while the stresses are 1,000,000 Pa (1 MPa). To remedy this situation, the COMSOL solvers internally rescale the variables so that a well-scaled system results.

The rescaling of the discretized linear system occurs before constraint handling. Assume that the degrees of freedom Ui are expressed terms of rescaled degrees of freedom according to the formula

where si are positive scale factors. Using a diagonal matrix S, the relation between U and is , and you can write the rescaled linear system as

Here, R is a diagonal matrix of positive scale factors chosen such that the rows in the matrix are of magnitude 1.