In certain cases, boundary integral methods are preferred for elliptic partial differential equations as opposed to finite difference methods. For instance, for solving the Poisson equation in a domain $D$$$\nabla^2 u = \phi_0 \tag{$\star$}$$with boundary condition $u = f$ on $\partial D$, instead of discretizing $(\star)$, I often see the boundary integral formulation being solved, i.e.,$$u(x) = \int_{\partial D} \sigma(s) \dfrac{\partial G(x,y(s))}{\partial n_y} ds + \int_{D} G(x,y) \phi_0(y) dy$$where the unknown density $\sigma(s)$ is obtained by enforcing the boundary condition, i.e.,$$\dfrac{\sigma(x)}2 + \int_{\partial D} \sigma(s) \dfrac{\partial G(x,y(s))}{\partial n_y} ds + \int_{D} G(x,y) \phi_0(y) dy = f(x) \tag{$\dagger$}$$What is the advantage of solving $(\dagger)$ instead of $(\star)$? Note that in $(\dagger)$, one integral is done over the entire domain, while the other is done over the boundary. Below are the list of questions, which I have:What dictates the choice of boundary integral method and finite difference method, i.e., under what circumstances are boundary integral methods preferred over finite difference?Is one more computationally accurate and/or easier and/or inexpensive than the other?Overall, I would like to have a comparison between boundary integral methods and finite difference methods highlighting the pros and cons of both the methods.
Finite difference equations versus boundary integral equations for elliptic pdes
finite difference;integral equations