Parallel solution of the modified porous medium equation

Abstract

The present study deals with the numerical solution of the modified porous medium equation when the solution is subject to some constraints. First of all, we use a change of variables, which leads to an evolutive problem where the nonlinear part is constituted by the derivative with respect to the time, of a diagonal increasing operator. Then, for the numerical solution we consider an implicit time marching scheme, which leads to the solution of a sequence of stationary problems. In fact each stationary problem is equivalent to a constrained minimization problem, and for Dirichlet and Dirichlet–Neumann boundary conditions, we show the existence and the uniqueness of the solution of our stationary constrained minimization problem. Moreover, classically, the solution of each stationary constrained problem can be characterized by the solution of a multivalued one. The spatial discretization of the previous problem leads to the solution of large scale algebraic multivalued systems. Then we analyze in a unified approach the convergence of the sequential and parallel relaxation projected methods. Finally we present the results of numerical experiments.