An edge-bubble stabilized finite element method for fourth-order parabolic problems

We develop an edge-bubble stabilized finite element method for fourth-order parabolic problems. The method begins with a non-conforming approach, in which C 0 basis functions are used to approximate the coarse scale of the bulk field. Continuity of function derivatives is enforced at element edges with Lagrange multipliers. The fine-scale bulk field is approximated with higher order edge-bubbles that are held fixed over time slabs, providing for static condensation and an elimination of the multipliers. The resulting formulation shares several common features with recent non-conforming approaches based on Nitscheʹs method, albeit with the important difference that stability terms follow automatically from the approximation to the fine scale. As an application, we consider the problem of plane Poiseuille flow for a second-gradient fluid. Convergence studies provided for the case of steady flow indicate synchronous rates of convergence in L 2 and H 1 error norms. Some new time-dependent results for the second-gradient theory are also provided.