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.