Implicit Taylor methods for parabolic problems with nonsmooth data and applications to optimal heat control

摘要

As a rule, parabolic problems with nonsmooth data show rapid changes of its solution or even possess solutions of reduced smoothness. While for smooth data various time integration methods, e.g. the trapezoidal rule or the Euler backwards scheme, work efficiently, but in case of jumps effects of high-frequency oscillations are observable over a long time horizon or steep changes are smeared out. Implicit Taylor methods (ITM), which are mostly applied in specific applications, like interval methods, but not commonly used for general cases, combine high accuracy with strong damping of unwanted oscillations. These properties make them a good choice in case of nonsmooth data. In the present paper ITM are investigated in detail for semi-discrete linear parabolic problems. In ITM at each time level a large-scale linear system has to be solved and preconditioned conjugate gradient methods (PCG) can efficiently be applied. Here adapted preconditioners are constructed, and tight spectral bounds are derived which are independent of the discretization parameters of the parabolic problem. As an important application ITM are considered in case of boundary heat control. Occurring control constraints are involved by means of penalty functions. To solve the completely discretized problem gradient-based numerical algorithms are used where the gradient of the objective is partially evaluated via discrete adjoints and partially by explicitly available terms corresponding to the penalties. Some test examples illustrate the efficiency of the considered algorithms.