skade eaction ffusion ne-dimensional System - ZIB

The nom whih KARDOS uses s a weighted L-no, giv by
whe
it
uAbsi
uRelMxi
\\ u
r-
||«
||^||o < f ^ i
uAfe, < ||u||o < uRelMxi
uRelM ||u
ReM RTOL o ,
uA6s ATOL * (O) 1 /
:i-7)
Scaling can be turned on or off. If the scaled error norm is used, the tolerances
RTOLi and ATOLi have to be seleced very carefully to reflect accurately
the scale of the problem. The tolrane ATOLi should indicate the absolute
value at which the i-th component is essentially insignifican. O the other
hand, the value RTOLi affects the relative accuracy of the i-th component
with respect to its maximal value in time. This control turns out to be quite
efficient and robust for a wid lass of problems. Howeer, it is cear that i
is not a univeral mthod.
To implement one time step, one elliptic proble has to be solved at each
tage of the Rosenbrock method. This is done by means of an adaptiv
ultilevel finite element method elaborated in recent years, see DEUFLHARD
et al. []. This method is an excellent tool to adapt the space discretization
for the current solution in uch a way, that a prescribed tolerance spaceTol
is achieved. KARDOS uses standard linear elements connected with local
error estimator of BABUSKA-HEINBOLDT [1] yp. We get at each tage,
discretizations of the form
B T v ) = n(v) e SnVvn e S i = 3 1.8)
where Sn denotes the space of al ntinuous, piecewise linear functions on
partition An of 0, satisfying the Dirichlet boundary conditions. In S% hey
vanish on the Dirichlet boundary. The equations only differ in the respective
righthand id and in the pecial boundary conditions taken in onsider
ation.