An error analysis of discrete Kirchhoff elements

The Discrete Kirchhoff Triangle (DKT) method for the biharmonic equation is analyzed in the discrete energy norm. The error is bounded by the best approximation of the Hessian by piecewise constants and the oscillation of the right-hand side, without additional regularity assumptions on the exact solution. This result implies first-order convergence of the classical DKT element and the analysis yields a canonical extension to three space dimensions with the same approximation properties. Residual-based a posteriori error estimates are derived. The analysis is formulated within a general framework for low-order nonconforming methods, which also applies to various classical elements and yields best-approximation results by constants. It is furthermore shown how known stable pairs for the planar Stokes system have discrete stream functions in discrete Kirchhoff spaces. This yields variants of the known schemes with positive definite formulations and pressure-robust error bounds.