Iterative methods for simulation of coupled engineering problems

摘要

In this work we intend to review the state-of-the-art of iterative methods for solving large sparse systems such as arising in coupled engineering problems using finite difference and finite element approximations. The solution of practical problems of mathematical physics ultimately relies on solving a system of partial derivative equations and this is only achieved by iterative numerical methods. Iterative solution methods proceed by adding successive corrections to some arbitrary initial approximation, but unfortunately these methods are very sensitive to specific features of the system to be solved. A procedure call preconditioning is possible but is not always used. We limit our presentation to a large class of systems defined by elliptic-parabolic mathematical models that represents the basis of the electromagnetic-thermal problems. The numerical models are obtained by the finite differences and finite element methods. The motivation is simple: for parabolic problems we use an explicit scheme for temporal discretization, and for elliptic problem we use the finite element method. As target example we use an electromagnetic-thermal coupled problem from electrical engineering.