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Optimal convergence analysis of mixed finite element methods for fourth-order elliptic and parabolic problems. (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:
65M12Stability and convergence of numerical methods (IVP of PDE)
65M15Error bounds (IVP of PDE)
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
65N15Error bounds (BVP of PDE)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
35J40Higher order elliptic equations, boundary value problems
35K30Higher order parabolic equations, initial value problems
65N12Stability and convergence of numerical methods (BVP of PDE)