# Tagged Questions

Referring to a time independent partial differential equation of the form $\nabla^2u=f$

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### What is wrong with my code for solving Poisson equation with one side Neumann boundary condition?

I wrote a Matlab code for solving 2D Poisson equation $u_{xx} + u_{yy} + f(x,y) = 0$ on $[a,b]\times [c,d]$ with neumann boundary condition on $x = b$ and the other boundary conditions are dirichlet,...
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### Is it reasonable to use B-splines to compute a radial symmetric electrostatic field?

When I use normal finite elements to compute the electrostatic potential, then the electric field itself is discontinuous at the interfaces of two cells. This impacts how electrons can be tracked ...
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### Discretize Poisson equation with derivative of delta function as source

Consider the PDE $$\frac{d^2}{dx^2} g(x) = \frac{d}{dx} \delta(x-x_0),$$ with $x, x_0 \in [0,1]$ and $g(0)=g(1)=0$. What is the best method to discretise the derivative of ...
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### Laplace's equation with periodic Dirichlet boundary conditions

Consider a Laplace's equation with Dirichlet boundary condition: ${\nabla ^2}\Phi = 0$ in a domain $D$ with given Dirichlet Boundary condition: $\Phi=\Phi_o$ at $\partial D$ (smooth, but not ...
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### Increasing V-cycles for constant Coarsest Grid Size and increasing Fine Grid size

I implemented geometric multigrid for $-\nabla^{2}=f$ where $f=\frac{3\pi^{2}}{4}sin \frac{\pi x}{2} sin \frac{\pi y}{2} sin \frac{\pi z}{2}$ on $\Omega \in [0,1]$ on a unit cube. Dirichlet boundaries ...
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### Visualization of Discontinuous Galerkin solution

Is there any simple way to visualise the solution of discontinous galerkin for poisson problem in matlab when monomials are used as basis function. ? Thanks alot.
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### How to solve a Poisson equation using the finite difference method when there is an object inside a domain?

I'm interested in solving an electrostatics problem in 2d case in some domain with a conductor placed inside the domain. From a mathematical point of view, I have to solve a Poisson equation with ...
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### 1D inhomogeneous Poisson PDE with Dirichlet BCs, slow convergence

For an assignment I have to implement a 1D Poisson PDE with inhomogeneous Dirichlet BC's $$\Delta_1 u = f, \quad u(a)=g(a), \: u(b) = g(b)$$ I have managed to make it work, but I am not seeing the ...
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### Simulation of Laplace Equation in 3-D with mixed BC of Dirichlet-Neumann

I am simulating a Mixed-Boundary value (Dirichlet-Neumann) problem using Finite Difference Method on a unit 3-D cube such that the left, lower, and front plane have $u=u(x,y,z)=1$ (Dirichlet) and ...
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### 2D Poisson Solver for Taylor Green Vortex Problem

I am trying to write a 2D Navier Stokes solver using an RK3 for time advancement in python. For debugging, I have converted the RK3 to an Euler step for simplicity. Checking my divergence for my ...
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### Boundary equations for constant right hand side in Poisson equation (FD)

I am setting up to solve a 2D Poisson equation $\nabla^2u = T$ by the finite difference method. I am using the multigrid method. On the coarsest mesh, where the grid size is 2x2, I want to set up ...
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### Particle mesh Ewald: recommended splitting into short and long range

Particle mesh Ewald method for acceleration of solving pairwise interaction by long range forces (electrostatic, gravitational ... ) seem to be very general and easy to implement. The basic principle ...
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### Solve poisson equation with Neumann b.c. (matlab or octave)

I'm trying to reconstruct an image given its Laplacian, which results in a Poisson equation and I'm using Neumann boundary conditions (derivative at boundary = 0). What I have is the laplacian (f, ...
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### The rate of convergence for finite difference methods for Poisson's equation with piecewise constant data

I am solving the following PDE; $$\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \rho,$$ where $\rho(0.5,0.5) = 2$ (zero elsewhere), $0\leq x,y\leq1$ and the ...
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### Applying Dirichlet boundary conditions to the Poisson equation with finite volume method

I would like to know how Dirichlet conditions are normally applied when using the finite volume method on a cell-centered non-uniform grid, My current implementation simply imposes the boundary ...
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### Given an SPD tridiagonal linear system, can we precompute so that any three indices can be linked in O(1) time?

Consider a symmetric positive definite tridiagonal linear system $$A x = b$$ where $A \in \mathbb{R}^{n \times n}$ and $b \in \mathbb{R}^n$. Given three indices $0 \le i < j < k < n$, if we ...
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### How can I calculate numerically an electrical potential distribution from an electric field distribution?

I want to calculate the unknown electrical potential distribution $\phi(x)$ (notice this is a function of $x$) from a known electric field distribution $\boldsymbol{E}(x)$ using the Poisson equation, ...
What is the theoretical convergence rate for an FFT Poison solver? I am solving a Poisson equation: $$\nabla^2 V_H(x, y, z) = -4\pi n(x, y, z)$$ with n(x, y, z) = {3\over\pi} ((x-1)^2 + (y-1)^2 + (...
I found this method in this book for solving the Poisson equation with error converging with $\mathcal{O}(h^4)$. However, when I try to implement it for the 1D equation $-u''(x)=f(x)$, it only ...