Stability, convergence and order of the extrapolations of the residual smoothing scheme in energy norm.

*(English)*Zbl 1230.65103The authors consider an abstract differential equation in a Hilbert space having in view application of the method of lines to parabolic equations discretized by spectral or finite element methods. The residual smoothing scheme consists in preconditioning the discrete time derivative of the explicit Euler method by a suitable operator. This, like the elliptic operator, is assumed to be self-adjoint and non-negative, further, it contains a free (stability) parameter. After introducing order relations for such operators and proving that the one-step operator has energy norm not exceeding 1, they investigate properties of Richardson extrapolation. The resulting operator is shown to be stable in the energy norm for sufficiently large values of the stability parameter and to assure convergence with any fixed order \(k\) in time.

Comparing with results of the alternating directions implicit (ADI) time method we can say that about fifty years ago cheaper ways of preconditioning parabolic equations were found and shown to converge, see Yanenko (1959), Bramble, D’yakonov and Samarskij – where the Russian papers all were translated to English [e.g., N. N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables. Berlin-Heidelberg-New York: Springer Verlag (1971; Zbl 0209.47103)]. True, the operators were imagined to arise from finite difference methods, and higher order was reached for \(d\)-dimensional parallelepipedons only. However, the stability theory of Samarskij was assuming essentially just the same properties like the present paper: self-adjointness, spectral equivalence of elliptic operator and preconditioner, and resulted convergence in energy norm (a German translation appeared in 1984 of his work from the seventies, see A. A. Samarskij [Theorie der Differenzenverfahren. Mathematik und ihre Anwendungen in Physik und Technik, Bd. 40. Leipzig: Akademische Verlagsgesellschaft Geest & Portig K.-G. (1984; Zbl 0543.65067)]).

Comparing with results of the alternating directions implicit (ADI) time method we can say that about fifty years ago cheaper ways of preconditioning parabolic equations were found and shown to converge, see Yanenko (1959), Bramble, D’yakonov and Samarskij – where the Russian papers all were translated to English [e.g., N. N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables. Berlin-Heidelberg-New York: Springer Verlag (1971; Zbl 0209.47103)]. True, the operators were imagined to arise from finite difference methods, and higher order was reached for \(d\)-dimensional parallelepipedons only. However, the stability theory of Samarskij was assuming essentially just the same properties like the present paper: self-adjointness, spectral equivalence of elliptic operator and preconditioner, and resulted convergence in energy norm (a German translation appeared in 1984 of his work from the seventies, see A. A. Samarskij [Theorie der Differenzenverfahren. Mathematik und ihre Anwendungen in Physik und Technik, Bd. 40. Leipzig: Akademische Verlagsgesellschaft Geest & Portig K.-G. (1984; Zbl 0543.65067)]).

Reviewer: Gisbert Stoyan (Budapest)

##### MSC:

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

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 |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35K20 | Initial-boundary value problems for second-order parabolic equations |

34G10 | Linear differential equations in abstract spaces |

65L05 | Numerical methods for initial value problems |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

65F08 | Preconditioners for iterative methods |