Approximating the Shape Operator with the Surface Hellan–Herrmann–Johnson Element

.We present a finite element technique for approximating the surface Hessian of a discrete scalar function on triangulated surfaces embedded in \(R³\), with or without boundary. We then extend the method to compute approximations of the full shape operator of the underlying surface using only the known discrete surface. The method is based on the Hellan–Herrmann–Johnson element and does not require any ad hoc modifications. Convergence is established provided the discrete surface satisfies a Lagrange interpolation property related to the exact surface. The convergence rate, in \(L^2\), for the shape operator approximation is \(O(h^m)\), where \(m ≥ 1\) is the polynomial degree of the surface, i.e., the method converges even for piecewise linear surface triangulations. For surfaces with boundary, some additional boundary data is needed to establish optimal convergence, e.g., boundary information about the surface normal vector or the curvature in the co-normal direction. Numerical examples are given on nontrivial surfaces that demonstrate our error estimates and the efficacy of the method.