Finite difference equations versus boundary integral equations for elliptic pdes

Question

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.

Answer 1

The question asked is too broad, in my opinion.

Assessment of BIE methods vs. finite difference methods (or other domain methods) require careful analysis of many different points. Among them

• is $D \subset \mathbb{R}^2$ or $D \subset \mathbb{R}^3$ or even $D \subset \mathbb{R}^N$ with $N>3$?

• is $D$ finite or infinite? is $D$ simply connected?

• which is the $\partial D$-surface to $D$-volume ratio compared to a domain diameter?

• how smooth is $\partial D$? is the analytical solution singular at the boundary or not?

• are you interested in an overall picture of the solution on $D$ ore you are looking for a very precise solution in just a few points?

• if the solution is singular at some point, are interested in obtaining a very good asymptotic approximation at those points?

This said, I will not go into summarizing BIE methods (this could be the subject of a book chapter!) but just add some random remarks.

1. For sure FD on $(\star)$ is easier than say collocation on $(\dagger)$: $G(x,y)$ is singular for $y\rightarrow x$, you have to master CPV and FP integration to correctly implement a BIE method.

2. With FD you compute an approximation of $u$ in $D$. With BIE you compute quantities on $\partial D$ and you have to recover values on $D$ by an integral representation.

3. FD give raise to sparse matrices, BIE methods to dense ones.

So there are specific applications (class of problems) in which BIE methods are superior to FD methods, and situations in which they are just terrible, but I don't think it is possible to summarize them in a short answer.