8
votes
Accepted
The rate of convergence for finite difference methods for Poisson's equation with piecewise constant data
Let's examine the one-dimensional three-point stencil case in detail,
because I think it's important to be clear just how this behaviour
arises, and what it means to set a point to a certain value in ...
- 11.4k
8
votes
Accepted
Solving Ax = b with sparse A and sparse b
When looking at the solution of your system, you will find that almost all entries of $x$ are nonzero although the right-hand side is "sparse". Hence, whatever algorithm you use, it'll have to visit ...
- 3,058
8
votes
Accepted
Solving Poisson equation with current BC using FEM
It is standard procedure for general case like that. You can enforce condition for electrode by Lagrange multipliers,
$$
L(u,\lambda) = \int_\Omega \sigma u_{,i} u_{,i} \, \textrm{d}V - \int_{\...
- 906
6
votes
Accepted
Jump condition for elliptic equation in standard finite element method
Preliminary remarks:
The natural jump condition is on the normal derivative, not on the full gradient.
The fact that the solution is continuous across the interface is part of the formulation of the ...
- 1,246
6
votes
Finite differences vs. elements: Accuracy and implementation
For a 6x6 grid, those are about the error differences I would expect from two different methods. You have to realize that a 6x6 grid is a very coarse grid, even for a simple problem like yours. As ...
- 4,567
6
votes
Accepted
How do I avoid divide-by-zero when solving the Poisson equation with Fourier transforms?
You are solving Poisson's equation:
$$\nabla^2 \Phi = 4\pi G \rho.$$
Notice that if $\Phi$ is a solution, then so is $\Phi+C$ for any constant $C$. Furthermore, $C$ will have no effect on your ...
- 16.2k
5
votes
Accepted
How to solve a Poisson equation using the finite difference method when there is an object inside a domain?
If you have simple, grid-aligned interior objects or accuracy is not crucial then stick with the method you have described. If you need to accurately represent arbitrary boundary shapes then you're ...
- 4,521
5
votes
1D inhomogeneous Poisson PDE with Dirichlet BCs, slow convergence
The answers so far seem to me to be completely off target. The boundary conditions you have are consistent with the r.h.s. of the equation and the matrix $L$, so they are not the problem.
Instead, ...
- 11.4k
5
votes
Poisson equation with Neumann boundary conditions
I think this happened because $L$ is a finite-difference operator that approximates your equation with some error (it is $\mathcal O(dx^2)$). To get small error your should increase ...
4
votes
explain the difference between 1D Poisson solvers
First, some notation for clarity. We're talking about solving $u_{xx}=f(x)$. So $u$ is the solution and $f$ is the right hand side.
What makes you think either of your solutions is or is not correct?
...
- 4,521
4
votes
Accepted
Is it necessary to use Poisson - Boltzmann equation if I only need to build electrostatic potential from a PQR file?
So Is it neccesary to use Poisson - Boltzmann equation if I only need to build electrostatic potential from a PQR file?
No. You can use Poisson.
Since you know the positions of each point charge, ...
- 207
4
votes
Accepted
Discrete Poisson Equation with Pure Neumann Boundary Conditions
Previous comments gave you good suggestions, I try to add some more.
Firstly, your example 2x2 really does not correspond to zero Neumann boundary conditions. In fact, one can show that your choice ...
- 1,206
4
votes
Discretize Poisson equation with derivative of delta function as source
As noted by others, this PDE is not well-posed in the Strong Form. However, you can take some function which asymtopically approximates the Dirac Delta and use that (where the error in the ...
- 11.5k
4
votes
Solving the Poisson equation with Neumann Boundary Conditions - Finite Difference, BiCGSTAB
One of the standard ways to handle singular problems like Poisson with Neumann boundary conditions which lead to a sinuglar problem $Ax=b$ is to make the obtained solution orthogonal to the kernel (...
- 254
4
votes
Jacobi iteration for finite difference: when to stop?
For solving systems you shouldn't be comparing the results between each iteration but rather computing the residual.
If you consider the matrix representation to be in the standard form:
$ A\cdot y ...
- 399
4
votes
Surface Normal for 2D Finite Element Method
As @BillGreen already said in one of the comments, the test function $v$ is actually zero on the boundary, so the whole boundary integral simply disappears.
If you want to understand why $v$ is zero, ...
- 50.2k
4
votes
Accepted
Iteratively solving 3D Poisson equation in MATLAB
The finite difference matrix for the Poisson equation is symmetric and positive definite.
So the preconditioned conjugate gradient algorithm is the iterative solver of choice for
this problem.
The ...
- 5,744
4
votes
Solving Poisson equation with current BC using FEM
Like @likask points out in the other answer, using Lagrange multipliers is one way to do this. The other is to recognize that if you have constraints of the form $\sum_i a_i U_i=0$, then you can ...
- 50.2k
4
votes
Accepted
Step3 in deal.II - Convergence of the mean
$h$ is a measure of the mesh size. In the example, they are using rectangular elements. For which a commonly used measure of the mesh size is the length of the largest diagonal.
Looking at the table ...
- 2,291
4
votes
Accepted
Integral of the Poisson Kernel
The tangent has periodic singularities, poles where it jumps from $+\infty$ to $-\infty$. At these points the inverse tangent will jump from $+\frac\pi2$ to $-\frac\pi2$. This is of course not ...
- 3,511
3
votes
Discrete Poisson Equation with Pure Neumann Boundary Conditions
Your difference stencil is wrong for nodes at the boundary (which in your 2x2 case happens to be every row). A simple way to see this is to realize that if you applied the Laplace operator to a ...
- 50.2k
3
votes
Discrete Poisson Equation with Pure Neumann Boundary Conditions
Pure Neumann boundary conditions give a non-unique solution. You will need to set some other condition to make it unique. The simplest would be to choose some solution value somewhere and set it to ...
- 542
3
votes
Recommendations on FEM software for implementing Nitsche's method on interfaces between matching meshes?
My suggestion is to just write your own code using PETSc instead of using an existing FE library. Parallel assembly/solve is the most complicated part of an FE code and PETSc takes care of it. The ...
- 1,769
3
votes
Choice of spaces for mixed formulation for Poisson Equation Or Darcy equation
Assuming you choose stable pairings of elements, the main consideration is which variable you are more interested in. For example, if you choose the $H(div),L^2$ formulation, then you have the choice ...
- 50.2k
3
votes
Accepted
3D Poisson equation, Fourier and Chebyshev
It is so trivial to pick a solution on a box domain for the Laplace equation. Just pick a function, say $\bar u(x,y,z)=x^2y^2\sin(z)$ (chosen in a way so that it isn't in your ansatz space), then ...
- 50.2k
3
votes
Poisson equation finite-difference with pure Neumann boundary conditions
I know this question is old but I found an error that I wanted to clear up in @DrHansGruber 's answer. It took me a few hours to spot the issue and I wanted to post the solution to save anyone else ...
- 73
3
votes
Boundary conditions in conjugate gradient method for poisson's equation
Neumann boundary conditions will show up in the right hand side of your problem, and so you can just start your iteration with any vector you want -- it doesn't have to satisfy any particular boundary ...
- 50.2k
3
votes
Solving Ax = b with sparse A and sparse b
It is unclear to me from your question whether the answer is of theoretical
or practical interest. I'll address both.
For this to be of practical importance, either your system would need
to have a ...
- 5,744
3
votes
Mixed formulation of the Poisson equation (FEM)
For this problem you cannot use arbitrary finite elements. The finite element spaces should satisfy the so-called discrete inf-sup condition. You did not mention what elements you used so I assume ...
- 1,832
3
votes
Linear solvers: How to deal with a singular system? (Poisson equation with Neumann boundary conditions)
Krylov solvers have no problem converging if the system is consistent, i.e., if the right-hand side is in the image of the system matrix $A$; see, e.g., Iterative Krylov Methods for Large Linear ...
- 3,058
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