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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 ...
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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 ...
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8 votes
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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_{\...
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6 votes
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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 ...
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  • 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 ...
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6 votes
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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 ...
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5 votes
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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 ...
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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, ...
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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 ...
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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? ...
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4 votes
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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, ...
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4 votes
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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 ...
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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 ...
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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 (...
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  • 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 ...
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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, ...
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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 ...
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  • 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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3 votes
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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 ...
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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 ...
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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 ...
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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 ...
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  • 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 ...
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  • 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 ...
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