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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_{\...
likask's user avatar
  • 906
8 votes
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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 ...
Nico Schlömer's user avatar
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 ...
David Ketcheson's user avatar
5 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 ...
user31765's user avatar
  • 101
4 votes
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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 ...
Bill Greene's user avatar
  • 6,064
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, ...
Wolfgang Bangerth's user avatar
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 ...
user3209427's user avatar
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 ...
Chris Rackauckas's user avatar
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 (...
VorKir's user avatar
  • 254
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 ...
Wolfgang Bangerth's user avatar
4 votes
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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 ...
Abdullah Ali Sivas's user avatar
4 votes
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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 ...
Lutz Lehmann's user avatar
  • 6,064
4 votes
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Does DCT diagonalize the FD discretisation of the Laplacian with Neumann boundary conditions?

The "issue" seems to have been that there is a discrepancy between the used transforms. The DCT transforms discussed in the linked paper are orthogonal, i.e. $U^{-1}=U^T$, where the even one ...
lightxbulb's user avatar
  • 2,122
4 votes

Convergence stall when solving 2D Poisson PDE with pure Neumann boundaries (finite differences)

For this equation, with the Neumann boundary conditions the problem becomes mathematically ill-posed; if there is a solution $\phi(x,y)$ then $\phi(x,y)+const$ is also a solution. This ill-posedness ...
Maxim Umansky's user avatar
4 votes

Poisson equation with discontinuous variable coefficient

First, some comments on discretization schemes for elliptic problems in general. There are some improvements you can make on the difference scheme that you've derived. The idealized problem is to ...
Daniel Shapero's user avatar
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 ...
knl's user avatar
  • 2,096
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 ...
Nico Schlömer's user avatar
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 ...
Bill Greene's user avatar
  • 6,064
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 ...
Wolfgang Bangerth's user avatar
3 votes

FV Discretization of source term in 2D Poisson Equation

The correct way is to average the source term which you can do easily in this case as you have a polynomial. In general you can do the average with a quadrature. For second order accuracy if the ...
cfdlab's user avatar
  • 3,028
3 votes

Tensor product representation for the 9-point finite difference approximations for the Poisson equation

This is too long for a comment, so I'll post an answer. If you start from the analytical 2D-Laplace operator, it naturally is already in a (sum of) tensor product form: $$ \Delta = \partial^2_x \...
davidhigh's user avatar
  • 3,127
3 votes
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Memory issues with iterative solvers

As you pointed out, your matrices are sparse which means that the number of non-zeros is small compared to the number of zeros. There are several formats to store such matrices, e.g. COO, CSC, CSR etc....
vydesaster's user avatar
3 votes

Discretization of Poisson's equation with 2d permittivity tensor

You have \begin{align} \nabla \cdot \left(\begin{pmatrix} a & b \\ c & d\end{pmatrix} \nabla \phi \right) &= \nabla\cdot\begin{pmatrix} a \partial_x\phi + b\partial_{y}\phi \\ c\partial_{x}...
lightxbulb's user avatar
  • 2,122
3 votes
Accepted

Solving Poisson's Equation with Periodic Boundary Conditions

Here's what I think the problem is. You have the equation: $$\Delta u(x) = f(x), \, x\in \Omega.$$ where $\Omega$ is a circle/torus. First you have some compatibility constraints you have to satisfy. ...
lightxbulb's user avatar
  • 2,122
2 votes
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Boundary treatment with higher order methods

Common Ghost cell treatment in CFD Ghost cell treatment is an art in CFD, we can't arbitrarily choose ghost cell for all the problems. Let's assume $0$ is boundary point and $-1$ is ghost point and $...
AGN's user avatar
  • 544
2 votes
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Explicitly including boundary points in a set of finite-difference equations

If you include the boundary condition directly in the matrix, you will only get the g value at the points where the boundary is prescribed. If we use 5 nodes with the following BCS: $$u_1=g_1$$ and $$...
BlaB's user avatar
  • 1,157
2 votes

Solving Poisson equation while suffering from the curse of dimensionality

The short answer is that you can't do this -- it's outside our computational power today. To explain why, think of just building the box itself, where you have one degree of freedom on each vertex. In ...
Wolfgang Bangerth's user avatar
2 votes
Accepted

Finite difference equations representing semilinear elliptic PDE

Since you don't end up with a linear system, you have to use a nonlinear root finder. The most common choice is Newton's method, because it converges quite quickly (if your initial guess is good ...
Lukas Bystricky's user avatar
2 votes
Accepted

FFT Poisson Solver for non-uniform grid

First, there are non-uniform mesh FFT variations that you could use without having to do the coordinate transformation. Second, the FFT is not easily applicable to the transformed problem. The reason ...
Wolfgang Bangerth's user avatar
2 votes
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Numerical error of a spectral-domain Poisson solver

The Laplacian is $\sim k^2$ for low wavenumbers; it is a general property and it does not depend of your discretization. You are solving the Poisson equation for two different RHS so the solutions are ...
floren's user avatar
  • 36

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