2.8 General Solving Procedure

The complete solving procedure for a time dependent problem has the following structure,

This yields values of $\mathbf{c}^n$ for discrete times $t_n=\sum_i \Delta t_i$. Automatic time step and mesh control are discussed in the next section. As the terminating condition (condition( $\mathbf{c}_h^{k,n}$) in the Solving Algorithm) for the Newton loop, the following inequalities are commonly used,

$$\Vert\mathbf{c}_h^{k+1,n}-\mathbf{c}_h^{k,n} \Vert\leq \eta_{abs},$$ (2.48)

$$\Vert\mathbf{c}_h^{k+1,n}-\mathbf{c}_h^{k,n} \Vert\leq \eta_{rel}\Vert\mathbf{c}_h^{k+1,n}\Vert,$$ (2.49)

$$\Vert\mathbf{R}(\mathbf{c}_h^{k+1,n})\Vert\leq \eta_f.$$ (2.50)

where $\eta_{abs}$, $\eta_{rel}$, and $\eta_f$ are given tolerances and $\Vert \Vert$ Euclidian norm.