Stability of difference schemes for parabolic equations with dynamical boundary conditions and conditions on conjugation.

*(English)*Zbl 1081.65089The authors prove stability results for difference schemes for parabolic equations. Most results are for one space dimension but a section for a weakly-parabolic equation in two dimensions is included. The authors emphasize the connection to abstract Cauchy problems in Hilbert spaces for both energy estimates and estimates on the maximal time step for stability of the difference schemes. In one space dimension, three equations are considered: a heat equation with concentrated capacity, a heat equation with a dynamical boundary condition (the time derivative on the unknown function is given at a boundary point) and a weakly parabolic equation is considered in both one and two dimensions. The results are verified with numerical examples.

Reviewer: Stefan Jakobsson (Stockholm)

##### MSC:

65M12 | Stability and convergence of numerical 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 |

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

65M15 | Error bounds 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 |

##### Keywords:

parabolic equations; finite difference schemes; stability; time step estimates; error bounds; numerical examples; Cauchy problems; Hilbert spaces; heat equation
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\textit{B. S. Jovanović} and \textit{L. G. Vulkov}, Appl. Math. Comput. 163, No. 2, 849--868 (2005; Zbl 1081.65089)

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