I have been working on a multigrid solution to a non-homogeneous Dirichlet boundary value problem.  However, the function goes to infinity on the boundary.  This causes numerical overflow errors to be raised in calculating the discretized derivatives.  Does anyone have a suggestion on how to overcome this problem?  I am not a mathemetician, but the Galerkin Method seems to be a way to transform the problem, but I am not sure.Here are the pde's I'm solving (I've not included the formulae for the drift and diffusion coefficients as they are also long):$\frac{1}{2} \omega_i^2 \sigma_{\omega_i}^2 \frac{\partial^2 f^i}{\partial \omega_i^2} + \frac{1}{2} \delta^2 \sigma_\delta^2 \frac{\partial^2 f^i}{\partial \delta^2} + \delta \omega_i \sigma_{\omega_i} \sigma_\delta \frac{\partial^2 f^i}{\partial \delta \partial \omega_i} + \delta \mu_\delta \frac{\partial f^i}{\partial \delta}$  $+ \omega_i \mu_{\omega_i} \frac{\partial f^i}{\omega_i}  - \rho f^i + (\omega^i)^{-\gamma_i} \left ( tan \left [ \pi(\delta - \frac{1}{2}) \right ] \right )^{1-\gamma_i}= 0$  $\frac{1}{2} \omega_i^2 \sigma_{\omega_i}^2 \frac{\partial^2 h^i}{\partial \omega_i^2} + \frac{1}{2} \delta^2 \sigma_\delta^2 \frac{\partial^2 h^i}{\partial \delta^2} + \delta \omega_i \sigma_{\omega_i} \sigma_\delta \frac{\partial^2 h^i}{\partial \delta \partial \omega_i} + \delta \mu_\delta \frac{\partial h^i}{\partial \delta}$  $+ \omega_i \mu_{\omega_i} \frac{\partial h^i}{\omega_i}  - \rho h^i + \omega^i(t)^{-\gamma_i} \left ( tan \left [ \pi(\delta(t) - \frac{1}{2}) \right ] \right )^{-\gamma_i}  = 0$  where$H_0(t) = exp \left ( -\int_0^t r(u) du \right) exp \left (- \int_0^t \frac{\sigma_D}{\xi(u)} du - \frac{1}{2} \int_0^t \left ( \frac{\sigma_D}{\xi(u)} \right)^2 du \right )$  
Infinite Function Value on Dirichlet Boundary
pde;finite difference;numerical analysis;multigrid;elliptic pde