Traditionally, preconditioners of numerical methods are devised using the structure of the algebraic systems. They often require information on input matrices, such as singular values, without using knowledge of the physical problems of the numerical methods model. This paper considers preconditioning and regularization methods for improving the convergence rate and accuracy of numerical solutions based on properties of the continuous equations, that is, the “physics” of the problem. One of the advantages of such an approach is that it does not depend on the discretization used, and thus a preconditioner may work equally well for a variety of grid options. In particular, both hyperbolic and elliptic partial differential equations are studied in the paper, with a number of examples presented.

One of the disadvantages of the approach is that the analysis cannot be generalized, so it has to be redone for different differential equations. Furthermore, even for a given equation, the analysis may also change, depending on other factors, such as whether it is an interior or exterior problem. Nonetheless, the approach proposed in the paper provides a valuable alternative to preconditioning should traditional approaches fail or be less effective.