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We obtain the spline recovery method on a $d$-dimensional simplex $T$ thatuses as information values and gradients of a function $f$ at the vertices of$T$ and is optimal for recovery of $f({\bf w})$ at every point ${\bf w}$ of anadmissible domain $P$ containing $T$ on the class $W^2(P)$ of twicedifferentiable functions on $P$ with uniformly bounded second order derivativesin any direction. If, in particular, every face of $T$ (of any dimension)contains its circumcenter, we can take $P=T$.  We also find the error function of the pointwise optimal method which turnsout to be a function in $W^2(P)$ with zero information. The error function is apiecewise quadratic $C^1$-function over a certain polyhedral partition and canbe considered as a multivariate analogue of the classical Euler spline$\phi_2$. The pointwise optimal method is a continuous spline of degree two(with some pieces of degree one) over the same partition.

Pointwise optimal multivariate spline method for recovery of twice differentiable functions on a simplex