absence of CFL like condition in numerical schemes of elliptic pde Numerical schemes for heat equation (parabolic $u_t=cu_{xx}$) and conservation laws(hyperbolic, $u_t+f(u)_x=0$ ) have restriction on the ratio of mesh size (i.e. $dx$ and $dt$ ratio).
Why don't we have such restrictions on the ratio of $dx$ and $dy$ when it comes to elliptic PDE like Laplace equation $u_{xx}+u_{yy}=0$?
 A: Let us talk here about second order partial differential equations (PDEs) with constant coefficients. The difference between elliptic PDEs and parabolic/hyperbolic PDEs is in their eigenvalues - elliptic PDEs have only complex eigenvalues. This means that they contain no real characteristics, i.e. no characteristic line, along which the information propagates (from region with information to region without information). Parabolic and hyperbolic PDEs contain always some real characteristics.
Meaning of this is, that if you numerically solve parabolic or hyperbolic PDEs (using a finite difference scheme for example) (let us consider independent variables $t$ and $x$), you can go from a region containing new information (typically from $t=0$, where initial condition $u(t=0,x)=u_0(x)$ is defined) to larger times $t>0$ (in direction of a characteristic line) and compute new time levels from the previous ones. In this way you numerically propagate information to new time levels. (In order to judge stability of the particular finite difference scheme in this case, one uses the von Neumann stability condition, which also yields the CFL condition. This condition tests, if an error which propagates with the solution along characteristic lines is damped or amplified by the particular finite difference scheme.).
In the elliptic PDE (let us consider independent variables $x$ and $y$ living in a domain $\Omega$), having no real characteristics, you cannot go from one part of the domain $\Omega$ to another, solving "unknown solution" from "already known part of the solution", because the solution at any point depends on the solution at any other point in $\Omega$. You have to solve the whole system at once, by forming a system of linear equations $Ax=b$ (after discretization), in which every solution points are interdependent. You get then by inversion the discretized solution at all points in $\Omega$ at once.
A: Elliptical PDEs can be defined by functions that do not have any characteristic lines/surfaces. That is, there are no functions $F(x)$ or $G(y)$ such that $u''(x,y)=F(x)$ or $u''(x,y)=G(y)$. 

Therefore, solutions to elliptical PDEs are dependent on both $x$ and $y$. Similarly, dx and dy must also share a dependence. This dependence concludes that solutions can not have discontinuities in their partial derivatives. For parabolic and hyperbolic PDEs, solutions can have discontinuities in their partial derivatives along the characteristics.
The difference, I believe, between elliptical and parabolic/hyperbolic PDEs, is that the dependencies, or restrictions, are built into the equations of elliptical PDEs; whereas the dependencies for parabolic/hyperbolic PDEs are artificially put in since the variables are independent of each other along the characteristics.
