Error analysis for implicit approximations to solutions to Cauchy problems.

*(English)*Zbl 0855.65102This paper is concerned with the error behaviour of some implicit approximations to the general Cauchy problem \(du(t)/dt + {\mathcal A} u(t) \ni 0\), \(u(0) = u_0 \in D({\mathcal A})\) where \(u : [0,T] \to {\mathcal H}\), being \(\mathcal H\) a Hilbert space and \(\mathcal A\) maximally monotone on \(\mathcal H\). The author considers the discrete approximations \(u_h\) given by Euler’s implicit method and a continuous \(u_\lambda\) obtained as the solution of: \((du_\lambda/dt) + {\mathcal A}_\lambda (u_\lambda) = 0\), \(u_\lambda(0) = u_0\) where \({\mathcal A}_\lambda\) is the Yoshida approximation of \(\mathcal A\).

After a brief revision of the main theoretical tools used in the paper, the author proves in Section 3 two theorems which provide estimates of \(u_\lambda - u\) first in a general setting and then in the case that \(\mathcal A\) is a subgradient. Next in section 4 similar results are proved in the case of a discrete approximation for either nonuniform and uniform grids. Moreover particularly interesting results are given for the case where \(\mathcal A\) is the sum of a subgradient and a monotone operator with some additional assumptions.

The paper ends with applications to several examples including linear nonnormal operators as the heat equation with a motion of the substance given by \({\mathcal A} u = -\Delta u + \nabla \cdot (vu)\) and nonlinear examples concerned with the Stefan problem. As remarked by the author these examples are intended to show the tightness of the estimates and the generality of the theoretical results proved in the paper.

After a brief revision of the main theoretical tools used in the paper, the author proves in Section 3 two theorems which provide estimates of \(u_\lambda - u\) first in a general setting and then in the case that \(\mathcal A\) is a subgradient. Next in section 4 similar results are proved in the case of a discrete approximation for either nonuniform and uniform grids. Moreover particularly interesting results are given for the case where \(\mathcal A\) is the sum of a subgradient and a monotone operator with some additional assumptions.

The paper ends with applications to several examples including linear nonnormal operators as the heat equation with a motion of the substance given by \({\mathcal A} u = -\Delta u + \nabla \cdot (vu)\) and nonlinear examples concerned with the Stefan problem. As remarked by the author these examples are intended to show the tightness of the estimates and the generality of the theoretical results proved in the paper.

Reviewer: M.Calvo (Zaragoza)

##### MSC:

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

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

34G20 | Nonlinear differential equations in abstract spaces |

65L05 | Numerical methods for initial value problems |

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

35K55 | Nonlinear parabolic equations |

80A22 | Stefan problems, phase changes, etc. |

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

35R35 | Free boundary problems for PDEs |