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.