*(English)*Zbl 1106.65081

Mixed finite element methods (FEMs) are considered on quasi-uniform rectangular decompositions of a spatial domain. The author applies a specific interpolation operator introduced by *V. Girault* and *P. A. Raviart* [Finite element methods for Navier-Stokes equations, Berlin: Springer (1986; Zbl 0585.65077)] to analyse the order of convergence in FEMs.

First, a linear scalar elliptic equation of fourth order is given including generalised Neumannn boundary conditions. To employ a mixed FEM, a corresponding weak formulation consisting of two equations is constructed. A Galerkin approach yields a linear system for the numerical approximation. A corresponding error estimate is proved using the specific interpolation operator in a Sobolev space.

Second, a linear scalar parabolic equation of fourth order, where generalised Neumann conditions in space and initial values in time arise, is discussed in a similar manner. Thereby, the derivative in time is discretised via the backward Euler scheme with equidistant step sizes. Consequently, the problem implies a sequence of weak formulations, which is solved by FEMs. A corresponding error bound is shown using the interpolation operator again. The author remarks that standard interpolation techniques can prove only lower orders of convergence.

Finally, a brief numerical example of an elliptic problem is presented, which confirms the predicted order of convergence via refining the mesh widths.

##### MSC:

65M12 | Stability and convergence of numerical methods (IVP of PDE) |

65M15 | Error bounds (IVP of PDE) |

65M60 | Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) |

65N15 | Error bounds (BVP of PDE) |

65N30 | Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) |

35J40 | Higher order elliptic equations, boundary value problems |

35K30 | Higher order parabolic equations, initial value problems |

65N12 | Stability and convergence of numerical methods (BVP of PDE) |