skade eaction ffusion ne-dimensional System - ZIB

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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.
To cntrol the pace dicretiztion, we restrict urselves to the firt stage of the Rosenbrock method which is exactly the semiimplicit Euler method applied to the system (1.2). According to a local principle, on each current grid an error estimation is computed. Therefore we consider the corresponding elliptic differental equation on each element, imposng the urrent solution of the Euler method UE as boundary conditions, and solve it approximately by a quadratic finite element method. The difference ek between the linear and quadratic approximation over the kth elent satisfies (e
Page 1 and 2: Tehnicl Repot TR 9-9 (July 193) Jen
Page 3 and 4: Contents Introducti efng a problem
Page 5: Yet this frm not suited to be imple
Page 9 and 10: haper efining roblem At first the u
Page 11 and 12: haper C o n d ng All specal command
Page 13 and 14: haper r o g e r only This chapter g
Page 15 and 16: int nonNegativity ontrol of the non
Page 17 and 18: eal spaceTolFac paceTolglobTol, san
Page 19 and 20: espping
Page 21 and 22: haper le In this chapte how how to
Page 23 and 24: eal valO, vail; real *fVal, int equ
Page 25 and 26: fVals[l] = 0.0; /* */ else ZIBStdOu
Page 27 and 28: seldraw 1 timesteppin timestep 0.00

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