Linear failure and cubic convergence for surface evolution with relaxed minimal deformation

The finite element scheme for relaxed minimal-deformation surface evolution prescribes normal velocity while allowing tangential mesh motion. We establish contrasting results for its original exact-integral formulation with the discontinuous geometric unit normals of the current elements. For linear elements, one fixed smooth torus, a tangential rigid-rotation field, and an admissible alternating mesh family give a positive lower bound for the uniform-in-time \(L^2\) error of the lifted flow map, on an interval where the discrete solution exists uniquely and remains nondegenerate. For cubic elements, every admissible mesh family converges for every smooth exact solution with smooth inverse on a fixed interval: the original discrete solution exists uniquely throughout that interval, and its lifted map error is at most \(C_T h^3\sqrt{1+|\log h|}\). The cubic proof first controls shape error by a signed-distance-weighted relative volume and then estimates weakly tangent map error. These results partially solve the low-degree convergence question; the quadratic case remains unresolved.