An adaptive multilevel approach to parabolic equations. II: Variable- order time discretization based on a multiplicative error correction.

*(English)*Zbl 0735.65066The paper is concerned with temporally homogeneous parabolic initial- boundary-value problems. For one space dimension the author implemented an extrapolated implicit Euler scheme together with an adaptive multilevel finite element method solver which handles the arising singularly perturbed elliptic subproblems. Very promising results were obtained [part I, ibid. 2, No. 4, 279-317 (1990; Zbl 0722.65055)], which led the author to try the same approach in two space dimensions since the theory developed is independent of space dimension.

The present paper tries to avoid several structural drawbacks of extrapolation methods by constructing a variable-order time discretization which corrects error approximations by multiplication in order to avoid differences. The structure of the arising elliptic subproblems (of only one kind at each time step, at most with different right hand sides) is analyzed.

The author gives numerical results for one space dimension; examples in two space dimensions are subjects for a forthcoming part of the paper. By the way, the rational and the restricted-denominator-PadĂ© approximations of \(e^{-z}\) found in the paper are highly interesting.

The present paper tries to avoid several structural drawbacks of extrapolation methods by constructing a variable-order time discretization which corrects error approximations by multiplication in order to avoid differences. The structure of the arising elliptic subproblems (of only one kind at each time step, at most with different right hand sides) is analyzed.

The author gives numerical results for one space dimension; examples in two space dimensions are subjects for a forthcoming part of the paper. By the way, the rational and the restricted-denominator-PadĂ© approximations of \(e^{-z}\) found in the paper are highly interesting.

Reviewer: E.Lanckau

##### MSC:

65M20 | Method of lines 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 |

65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |

35K15 | Initial value problems for second-order parabolic equations |