How can the Poisson equation be numerically solved when its matrix is ​ill-conditioned in nature?

Question

In the numerical solution of Poisson partial differential equation PPD by the method of finite differences FDM, the linear system of algebraic equations that is usually met: Ax = b. If the PPDE matrix A itself is inherently ill-conditioned, the linear system of the Ax = b equations is also ill-conditioned. The solution exists but is very difficult to find. The question is what is the least difficult way to find the solution, especially if the number of equations is large?

Answer 1

I would first ask: why in this case is the solution difficult to find?

Typically a discretization of the Laplace operator will be semi-definite (with kernel equal to the number of connected components in your domain) and poses little numerical difficulty for either direct solvers (such as sparse Cholesky; add a tiny diagonal shift to make the matrix positive-definite) or indirect solvers (even basic Jacobi iterations typically converge very quickly).

If the matrix is ill-conditioned because you have a huge grid of tiny elements, you can try multigrid methods. But this shouldn't be necessary unless you truly have a large number of equations (e.g. more than a million).