How can I generalize the approach of double mesh principle discussed in the post to 2-D problems like two asset option pricing problem $$ \begin{aligned} \frac{\partial u}{\partial t}=&-\frac{\sigma_{1}^{2} x^{2}}{2} \frac{\partial^{2} u}{\partial x^{2}}-\frac{\sigma_{2}^{2} y^{2}}{2} \frac{\partial^{2} u}{\partial y^{2}}-\rho \sigma_{1} \sigma_{2} x y \frac{\partial^{2} u}{\partial x \partial y}-r x \frac{\partial u}{\partial x}-r y \frac{\partial u}{\partial y}+r u \\ & \text { for }(x, y, t) \in(0, \infty) \times(0, \infty) \times[0, T) \end{aligned} $$ with the final payoff condition $u(x, y, T)=\max(K_1-x,K_2-y,0)$ at expiry $T$.
Here, I have the numerical solution $U_{m_1,m_2}^n\sim u(x_{m_1},y_{m_2},t_n),$ where, $x_{m_1}$ and $y_{m_2}$ are spatial points and $t_n$ is the temporal point, after a standard (uniform gridding) finite difference method. To facilitate the numerical method, conventional linear boundary conditions $u(x,y_{min},t)=u(x_{min},y,t)=0,u(x_{max},y,t)=x_{max}$ and, $u(x,y_{max},t)=y_{max}$ have been used.
Further, I am not interested in the error along a spatial line by fixing other spatial variable, Instead, I need the error along the plane at an instant $t=t_0$. Different types of errors are available for me by making use of different matrix norms, but I couldn't find theoretical ROC (rate of convergence).
Any help will be greatly appreciated. Thanks in advance.