# Fast method for evaluating at a single point the solution to a boundary value problem for a linear second-order elliptic PDE?

I have a PDE of the form $\Delta q(z) + \vec F(z)\cdot\nabla q(z) + G(z)=0$, for $z\in\mathbb{R}^n$, with boundary condition $\nabla q(z)\rightarrow 0$ as $z\rightarrow\infty$. If I only need to compute $\nabla q$ at a single point $z_0$, and only to a low precision, what's the fastest way to do that?

The only method I've come across which may address this is the method of random walks on balls, which does not seem to have gotten much traction in the literature. Are there any others?