Stability and convergence in numerical analysis. III: Linear investigation of nonlinear stability.

*(English)*Zbl 0695.65042Summary: In a previous paper [Numerical treatment of differential equations, Sel. Pap. 4th Int. Semin., NUMDIFF, Halle-Wittenberg/ GDR 1987, Teubner-Texte Math. 104, 216-226 (1988; Zbl 0679.65039)] we showed that several standard definitions of stability of nonlinear discretizations are so strong that they classify as unstable a number of useful discretizations. Then a weaker definition was introduced which, however, was powerful enough to imply, together with consistency, the existence and convergence of the discrete solutions.

In this paper we prove that, for smooth discretizations, stability in the new sense is equivalent to stability of its linearization around the theoretical solution. This fact does not imply that schemes with stable linearizations are automatically useful, due to the appearance of so- called stability thresholds. The abstract ideas introduced are applied to a concrete finite-element example, with a view to assessing the advantages of the new approach.

In this paper we prove that, for smooth discretizations, stability in the new sense is equivalent to stability of its linearization around the theoretical solution. This fact does not imply that schemes with stable linearizations are automatically useful, due to the appearance of so- called stability thresholds. The abstract ideas introduced are applied to a concrete finite-element example, with a view to assessing the advantages of the new approach.

##### MSC:

65J15 | Numerical solutions to equations with nonlinear operators (do not use 65Hxx) |

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

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

47J05 | Equations involving nonlinear operators (general) |

34B15 | Nonlinear boundary value problems for ordinary differential equations |