Residuals

Tolerances

Equation tolerances are described in terms of absolute and relative quantities:

tolerance       1e-6;
relTol          0.1;

If the equation initial residual satisfies either of the specified values, the system of equations are assumed solved and will not be evolved.

Calculation

The residual calculation is solver-specific. However, the general approach follows:

For a matrix system

$$\mat{A} \vec{x} = \vec{b},$$

the residual is defined as

$$\vec{r} = \vec{b} - \mat{A} \vec{x}.$$

We then apply residual scaling using the following normalisation procedure:

$$n = \sum \left( \mag{\mat{A}\vec{x} - \mat{A}\av{\vec{x}}} + \mag{\vec{b} - \mat{A}\av{\vec{x}}} \right)$$

where $\av{\vec{x}}$ is the average of the solution vector. The scaled residual is finally given by:

$$r = \frac{1}{n} \sum \mag{\vec{b} - \mat{A} \vec{x}}.$$

This form leads to a normalised residual of 1 for uniform systems, i.e. where $\vec{x} = \av{\vec{x}}$. However, this also shows that if the initial solution changes, e.g. using non-uniform conditions, the normalisation also changes, leading to a different convergence history.