The $C^1$-approximation of the level surfaces of functions defined on irregular meshes

We consider the problem of interpolating the level surfaces of functions in some classes (Lipschitz functions, continuously differentiable functions, functions whose gradient satisfies the Hölder condition, and twice continuously differentiable functions) given their values at the nodes of irregular meshes. We derive geometric conditions on the triangulations of a sequence of finite collections of points which guarantee that the gradients of piecewise linear approximations converge. We illustrate the sharpness of these conditions with Schwartz's example. We propose a method for approximating level surfaces which guarantees $C^1$-convergence without any restrictions on the location of nodes.