are purely computational aspects which will be exploited elsewhere.

Note that we have constructed an approximation of the squared metric $d^2$. The metric $d$ is then approximated naturally by the squareroot of the approximation of the squared metric, i.e. we consider the function $$x\rightarrow d_{Nk}(x,x^k):=\sqrt{d^2_{Nk}(x,x^k)}$$ to be the approximation of the metric function $x\rightarrow d(x,x^k)$.

Construction of $H^{s,p}$-approximations
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We refine the construction of the preceding section by construction of an approximation which solves not only , (or the set of equations , with boundary conditions ), but also all multivariate derivatives of up to a given order $m$ at the interpolation points. It turns out then that these polynomials are $H^{s,p}$-approximations for $s\leq m$. The approximation is constructed recursively again. For a multiindex $\beta$ of order $|\beta|=m\geq 3$ we denote the approximations of order $ d^2_{M(\beta_m)^{n,N}}$ or just $d^2_{M(\beta_m)}$ if we do not want to refer to the number of interpolation points $N$ and the dimension of the problem $n$ explicitly. The choice of the mesh is free again (in principle). We just assume that a set $\left\lbrace x_1,\cdots ,x_N \right\rbrace$ of interpolation points is given. Again we may construct functions $x\rightarrow d^2_{M(\beta)0}(x,y)$, $x\rightarrow d^2_{M(\beta)0}(x,x^1)$,..., and $x\rightarrow d^2_{M(\beta)0}(x,x^N)$. We 