Numerical solution techniques for 6th order PDE's

[Hologram0110]

Hello everyone,

I've encountered a bit of an interesting problem; a 6th order PDE in 2D + time resulting from a phase-field type physics model (mathematically it comes from a mass conservation equation (two orders)coupled to the Euler-Lagrange equation for variational calculus when the function depends on x, x' and x'' (4 orders)). With normal phase field, (4th order) we are able to routinely solve them splitting it into two coupled second order equations. I think that in theory, I should be able to do the same here with three coupled second order equations. However, my attempts at numerical solutions with Comsol (finite-element) have not been successful. Does anyone have any resources that might be of assistance? Is there anything extra to know about 6th order equation compared to a 4th order equation?

In theory, a finite-element package with C2 continuity should be able to do it without splitting, but that appears to be an extremely exotic request. Is there a way of creating C2 continuous elements with coupled Lagrange elements in a robust way?

I think similar equations might occur in plate or membrane bending problems, but I haven't actually found anything.

 

[Hologram0110]

I guess I should explain what I'm looking for a bit more explicitly.

Mass conservation
$$\frac{dz}{dt}=\nabla \cdot c \nabla \mu$$
Chemical potential (Euler-Lagrange equation)
$$\mu = \frac{\partial E}{\partial z} - \frac{\partial }{\partial x} \frac{\partial E}{\partial z_x} -\frac{\partial }{\partial y} \frac{\partial E}{\partial z_y} + \frac{\partial ^2 }{\partial x^2} \frac{\partial E}{\partial z_{xx}} + \frac{\partial ^2 }{\partial x \partial y} \frac{\partial E}{\partial z_{xy}} + \frac{\partial ^2 }{\partial y^2} \frac{\partial E}{\partial z_{yy}} $$
Energy
$$E=f \left( z(x,y), z_x(x,y), z_y(x,y), z_{xx}(x,y), z_{xy}(x,y), z_{yy}(x,y) \right) $$

I'm looking for information requirements to solve such a system (preferably with finite-element). For example, I assume that C2 continuity between elements would do it but this is hard to find. Would a C1 (Argyris)+ C0 (Lagrange) between elements do it? What about 3 C0 (Lagrange) elements?

Are there any order requirements on the order internal to each element? I'm guessing that is based on the number of derivatives I take of each term