On the solution of non-linear diffusion equation

Solution of the diffusion equation is usually performed with the finite element discretization for the spatial elliptic part of the equation and the time dependency is integrated via some difference scheme, often the trapezoidal rule (Crank-Nicolson) or the unconditionally stable semi-implicit two-step algorithm of Lees. A common procedure is to use Picard’s iteration with the trapezoidal rule. However, in highly non-linear problems the convergence of Picard’s iteration is untolerably slow. A simple remedy is to use consistent linearization and Newton’s method. For a certain class of non-linear constitutive models the consistent Jacobian matrix is unsymmetric. This paper discusses the use of the symmetric part of the Jacobian matrix and a combined Newton-type iteration scheme. Numerical results of highly non-linear diffusion problems are shown. Also a note concerning temporal discretization is given.