Hello,
As part of an investigation into adaptive and moving mesh techniques, I have been exploring the possible use of fenics to solve Winslow’s equations for (x,y) on a domain \xi\in[0, 1], \eta \in [0, 1]:
\alpha x_{\xi\xi} - 2\beta x_{\xi\eta} + \gamma x_{\eta\eta} = -[D_{\xi}(x,y)y_{\eta} - D_{\eta}(x,y)y_{\xi}]\dfrac{J}{D(x,y)}
\alpha y_{\xi\xi} - 2\beta y_{\xi\eta} + \gamma y_{\eta\eta} = -[D_{\xi}(x,y)x_{\eta} - D_{\eta}(x,y)x_{\xi}]\dfrac{J}{D(x,y)}
where
\alpha = x_{\eta}^2 + y_{\eta}^2
\beta = x_{\xi}x_{\eta} + y_{\xi}y_{\eta}
\gamma = x_{\xi}^2 + y_{\xi}^2
and J is the Jacobian determinant J = x_{\xi}y_{\eta} - x_{\eta}y_{\xi} , and where u_{v} = \dfrac{\partial u}{\partial v} and u_{vw} = \dfrac{\partial^2 u}{\partial v \partial w}.
The above are the inverse to the laplace equations
\nabla\cdot(D(x,y)\nabla\xi) = 0 and \nabla\cdot(D(x,y)\nabla \eta) = 0
giving a mapping between (x,y) and (\xi, \eta).
While the original laplace equations for \xi and \eta are trivially easy to implement using fenics, I am honestly struggling to obtain a weak form for this set of ‘inverse laplace’ equations. Particularly with dealing with the second-order derivatives when they are also multiplied by the first-order terms \alpha, \beta, \gamma. Most papers that use this method solves it using finite differences instead.
If anybody could provide some guidance that would be greatly appreciated!
Thanks,
Tim