n the previous section
we argued that we must have implicit form of the penalty
term:
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(Implicit equation with penalty term)
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We arrived to a non-linear equation. The non-linear part
is
where the scalar products
take both positive and negative values. We do not have convexity. Then, to
have an efficient way of finding a solution, we must have at least a
continuous matrix of first derivatives. Hence, we pick up the idea of
substitution
already introduced in the section
(
Time discretization
of penalty term
). The function
must have the following
properties:
It suffices to
set
for a parameter
.
Taking
substantially above 1 introduces uneven weights to values in the area
and, thus, contributes to stiffness. However, taking
significantly above 1 also increases smoothness and provides access to high
order calculational techniques.
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