Numerical algorithms for solving large systems

We saw in the article that the discretization of stationary partial differential equations led to the solution of high-dimensional linear systems with hollow matrices. Similarly, the discretization of evolutionary partial differential equations by implicit schemes (article ) also leads to the resolution of linear systems with the same characteristics. In view of this specificity, the inversion of matrices resulting from the discretization of partial differential equations is becoming an increasing concern in the field of numerical simulation, and is consequently very tricky, particularly in view of the poor conditioning of these matrices. This aspect is highly dependent on the applications dealt with, and it is out of the question to give a universal answer to this problem. That's why, in this article, we will review different methods for solving such systems, in an attempt to identify the most efficient algorithms.

In the case of the numerical solution of a non-linear partial differential equation, we have to solve a non-linear algebraic system; the solution of such a system will be carried out by an iterative method of Newton's type , which requires, at each iteration, a linearization of the application considered around the current point and the solution of a linear system; the study of the convergence of this type of method is far from trivial and theoretical results guaranteeing the convergence of the method are only established in particular situations. If the partial differential equation is linear, we'll have to solve a linear system which, in theory, seems simpler; however, there are still numerical difficulties in determining the approximate solution. In this presentation, we shall confine ourselves to the linear case.