4

By itself, Schwarz methods are stationary iterations just like Jacobi, Gauss-Seidel, or SOR. They converge to the solution, but often quite slowly. But, like any other stationary method, one iteration (or a fixed, small number of iterations) can also used as a preconditioner in Krylov-space methods. In other words, the distinction you are asking about is the ...


3

Here is at least an idea, whether it works is a different question. Let's say you sort unknowns so that you have the ones in the interior of the domain first, and then all those at the boundary. Then the matrix that corresponds to your problem decomposes in the following way: $$ A = \begin{pmatrix} A^{\circ\circ} & B^{\circ\partial} \\ C^{\...


2

In general, the appropriate preconditioners for elliptic problems such as yours are multigrid methods. In this 1d case, however, the simplest discretizations lead to tri-diagonal matrices and in that case the Thomas algorithm can be used to solve the problem directly without too much trouble. So you don't even need a preconditioner if you use a three-point ...


2

If you don't care too much about which $b$ you're working with (say just for linear algebra), then you can use the Method of Manufactured Solutions (MMS) in Linear Algebra much like we differential equations geeks would for our problems to start with a known exact solution, $x^*$, derive a $b$, and then apply your solver to see how decent an $x$ you can ...


2

The naive idea doesn't work immediately. The solutions of $$ \min \Bigg\|\begin{pmatrix} A \\ I \end{pmatrix} x - \begin{pmatrix} b_0\\b_1 \end{pmatrix} \Bigg\|$$ and $$ \min \Bigg\|\begin{pmatrix} P^{-1}A \\ I \end{pmatrix} x - \begin{pmatrix} P^{-1}b_0\\b_1 \end{pmatrix} \Bigg\|$$ are different, in general (unless $P$ is orthogonal).


2

As others have pointed out, (algebraic) multigrid can actually be a good preconditioner in this scenario. Below is a proof-of-concept implementation with scikit-fem and pyamg. It shows that pyamg's preconditioner makes the number of GMRES iterations more or less independent of the number of unknowns. I only tested this with a mesh of a few thousand unknowns. ...


1

If you are not enforcing any boundary conditions, you should remove the nullspace of the problem (to make sure that there is a unique solution, MatSetNullSpace can be used to achieve this in PETSc) and then use multigrid on the problem (for example, PCMG of PETSc). What surprises me is that there are n+1 zero eigenvalues. I would expect only one. Are you ...


1

I think there is a bit of jargon confusion here. So I will emphasize some words and the context I am using them. In NLA, when I attempt to solve a problem (they usually come from discretizations of PDEs), I know that the problem is well-posed, i.e. has a unique solution, beforehand. The problem may be ill-conditioned (opposed to well-conditioned) and that ...


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