I've just been introduced to the Poisson's equation. I've never had the need to dealt with PDE, so I'm a bit lost.Apparently we can compute an approximate solution of the Poisson's equation$$\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} u(x, y) = f(x, y)$$by discretizing the 2D Poisson's equation using finite differences. Here's a picture of that discretization taken from the website I've just linked you to:According to that website, we have:The above linear equation relating $U(i,j)$ and the value at its neighbors (indicated by the blue stencil) must hold for $1 <= i,j <= n$, giving us $N=n^2$ equations in $N$ unknowns. where by above linear equation I guess they are referring to$$4u_{i,j}+u_{i+1,j}+u_{i1,j}+u_{i,j+1}+u_{i,j1} = b_{i,j}$$$$1 \leq i, j \leq n$$When $(i,j)$ is adjacent to a boundary ($i=1$ or $j=1$ or $i=n$ or $j=n$), one or more of the $U(i+-1, j+-1)$ values is on the boundary and therefore $0$. $$b(i,j) = -f(i*h,j*h)*h^2$$ the scaled value of the right-hand-side function $f(x,y)$ at the corresponding grid point $(i,j)$.QuestionsWhere does the $-4$ in front of $u_{i,j}$ in the equation above comes from?What's $b(i, j)$? Why is it equal to $4u_{i,j}+u_{i+1,j}+u_{i1,j}+u_{i,j+1}+u_{i,j1}$? I mean, I don't understand where does it come from. Usually $b$ refers to a right-hand side, but...Why do we have $b(i,j) = -f(i*h,j*h)*h^2$?
Use Finite Difference Discretization to find approximate solution to the Poisson's equation
finite difference;numerical analysis;discretization;poisson