A finite element algorithm is constructed for a material coordinate formulation of the equations of sea-ice dynamics, using quadratic elements and fully implicit time steps. The material coordinate description allows the nodes of a fixed finite element mesh to define the same material elements as time proceeds, which avoids interpolation of nodal values on a changing spatial mesh as the pack evolves; quadratic elements preserve continuity of second derivatives, and this time stepping is stable and accurate in standard problems. We study a wind-driven pack with two free boundary sections, using a material description and fully implicit time steps, with a smoothed transition to zero stress in diverging flow, which significantly extends the time over which a stable solution is obtained. Stability and accuracy of the present algorithm is demonstrated by comparison with a class of exact solutions to other specific problems.