This work concerns the development of numerical methods for fourth-order partial differential equations over arbitrary domains. Finite-element methods based on similar equations involving fourth-order differential operators typically rely on C1-continuous basis functions or a mixed approach, both of which entail certain implementational difficulties. The objective of this thesis is to advance a relatively inexpensive, non-conforming method based on C⁰-continuous basis functions.

The method is applied to discretize a recent second-gradient theory for the flow of incompressible fluids. The new theory gives rise to a flow equation involving higherorder gradients of the velocity field and introduces an accompanying length scale and boundary conditions. We first develop the variational form of the method and establish consistency. The method weakly enforces continuity of the vorticity, traction, and hypertraction across interelement boundaries. Stabilization is achieved via Nitsche's method. Further, pressure stabilization scales with the higher-order moduli, so that the classical formulation is recovered as a particular limit. The numerical method is verified for the problem of steady, plane Poiseuille flow. We then provide several numerical examples illustrating the robustness of the method and contrasting the predictions to those provided by classical Navier-Stokes theory. We also prove the performance of the newly developed method by applying it to the Cahn-Hilliard equation, which is a nonlinear and fourth-order problem, for phase separation (or spinodal decomposition).

Further, numerical studies of turbulent flow based on a generalized Navier-Stokes-α theory are performed. The theory introduces two length scales α and β to describe turbulence in the inertial and dissipation ranges. We develop a pseudospectral method for the generalized Navier-Stokes-α model. The influence of the length scales are investigated through the turbulence energy spectrum.

Finally, an edge-bubble stabilized finite element method for fourth-order parabolic problems is proposed. The method begins with a non-conforming approach, in which C₀ 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 nonconforming approaches based on Nitsche's method, albeit with the important difference that stability terms follow automatically from the approximation to the fine scale. A relatively simple study of time-stepping algorithms leads to an implicit method as being the most practical choice. 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 L² and H ¹ error norms. Some new time-dependent results for the second-gradient theory are also provided.