Stable discretizations of the Cahn-Hilliard-Gurtin equations.

*(English)*Zbl 1172.65050The Cahn-Hilliard-Gurtin equations with a polynomial nonlinearity and with periodic boundary conditions are considered. Space and time discretizations are studied. The problem of numerical stability as defined by A. R. Humphries and A. M. Stuart [SIAM Rev. 36, No. 2, 226–257 (1994; Zbl 0807.65091)] is investigated. This means to find out if the discretization inherits the dynamical properties of the continuous problem.

A Galerkin method, which includes the case of conforming finite elements, is considered for space discretization. A convergence result is obtained. The numerical stability of the fully discrete scheme obtained by applying the Euler implicit method to the space semi-discrete problem is analysed. It is demonstrated that the fully discrete problem is unconditionally stable and the convergence to the solution of the continuous problem is proved. Numerical 1D experiments are presented.

A Galerkin method, which includes the case of conforming finite elements, is considered for space discretization. A convergence result is obtained. The numerical stability of the fully discrete scheme obtained by applying the Euler implicit method to the space semi-discrete problem is analysed. It is demonstrated that the fully discrete problem is unconditionally stable and the convergence to the solution of the continuous problem is proved. Numerical 1D experiments are presented.

Reviewer: Viorel Arnăutu (Iaşi)

##### MSC:

65M12 | Stability and convergence of numerical methods 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 |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

35Q53 | KdV equations (Korteweg-de Vries equations) |