Abstract: We present error bounds for the approximation from the spaces of multivariate piecewise polynomials admitting stable local bases. In particular, the bounds apply to the spaces of smooth finite elements in n variables. In addition, in the case of a space of bivariate quintic C1 piecewise polynomials we discuss its stable splitting into a subspace satisfying homogeneous boundary conditions and its complement. These results are used by K. Böhmer [3] in his finite element method for general fully nonlinear elliptic differential equations of second order.

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