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An adaptive multilevel approach to parabolic equations. I: General theory and 1D implementation. (English) Zbl 0722.65055
Nearly all approaches for the numerical solution of parabolic equations separate the discretization of time and space. One usually assumes one “outer” discretization to be carried out first, which leads to a semidiscrete problem. After that one continues to perform the second “inner” discretization, ending up with a fully discrete scheme. Here the discretization “first time then space” - classically known as method of Rothe - is chosen. With that sequence it is practicable to perform a multilevel matching of the inner and the outer discretization, which involves solution of the inner problem up to an accuracy matched with the accuracy of the outer problem.
The top levels consist in a low-order single-step discretization in time with extrapolation in Hilbert space, which yields variable time steps and variable orders controlled by the problem up to a given accuracy. The occurring elliptic problems are solved by multilevel methods, which produce the adequate individual space meshes in order to assure an accuracy required by the time discretization. In an error analysis the main result carefully traces the role of inconsistent and nonsmooth initial data. The estimates are sharp (in a certain sense).
The elliptic solver has been specified for the spatially one-dimensional case. Challenging numerical examples (rise of a traveling wave from a trivial solution, two counter-traveling waves, i.e. an example due to M. Bietermann, I. Babuška [J. Comput. Phys. 63, 33-66 (1986; Zbl 0596.65084)], point source) are included.

MSC:
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
Software:
Kaskade7
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