What are some of the "preferred" approaches for solving nonlinear PDE's? linear PDE's are simple enough, just use finite differences and arrive at some Ax=b system. However, I'm stuck on nonlinear PDE's. What are some preferred methods / best practices? Obviously it depends on the given PDE but is there any general way of approaching such problems?

As an example, consider the nonlinear ODE in pic related, with u=u(x,y). For simplicity, let omega be the unit square [0,1] x [0,1]. How would one go about solving such a nonlinear PDE numerically?