The paper constitutes a direct continuation of papers with the same main title [ibid. 2, 279-317 (1990; Zbl 0722.65055); 3, 93-122 (1991; Zbl 0735.65066)]. We remember: nearly all approaches for the numerical solution of parabolic equations separate the discretization of time and space — “first time then space”.
Hence, a problem arises: the remaining elliptic subproblems are singularly perturbed resulting from the time step. Hence standard adaptive finite element solvers run into difficulties for small time steps occurring in transient phases. The method to overcome this difficulty is preconditioning. Two devices have to be constructed: error estimator and linear solver. Using a multilevel iteration, a proper preconditioner is needed, this is the key to the error estimator, too.
Here the conceptual base is the preconditioner presented recently by J. H. Bramble, J. E. Pasciak, and J. Xu [Math. Comp. 55, 1-22 (1990; Zbl 0703.65076)], extended to highly nonuniform meshes by H. Yserentant [Num. Math. 58, No. 2, 163-184 (1990; Zbl 0708.65103)]. Now a two-dimensional version of the algorithm is given. In order to give applications to real life problems, which combines the difficulties of complex problem geometry, discontinuous coefficients and others, the author presents the solution of the bioheat-transfer equation in the framework of hyperthermia. The method in principle allows an on-line computation on a workstation.