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

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1answer
86 views

Solve 3-D Heat equation with Neumann boundaries

I want to solve the Poisson PDE for heat flow in a 3-D solid cube with given dimensions $x$, $y$, and $z$: $$\rho C\frac{\partial T}{\partial t} = k \Delta T$$ The cube is irradiated with a constant ...
3
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0answers
68 views

Unstable convergence of a Poisson equation

What could be the reason that the solution of a Poisson equation is smooth when obtained by an iterative solver, only if the maximum residual is set to a high value (e.g. 0.1)? When the maximum ...
5
votes
1answer
86 views

Numerical error of a spectral-domain Poisson solver

In $\mathbb{R}^n$, I would like to solve a Poisson equation (given $f$, solve for $u$): $$\nabla^2 u = f$$ assuming Neumann boundary condition (i.e. $\partial u = 0$ at boundaries). I solved it in ...
0
votes
1answer
90 views

Discontinuous Galerkin - Inhomogeneous Dirichlet B.C. for 1D Poisson Equation

I am trying to get some code working for the 1D Poisson equation using the textbook: Nodal Discontinuous Galerkin Methods Algorithms, Analysis, and Applications. I use the following formulation (for ...
1
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1answer
86 views

FFT Poisson Solver for non-uniform grid

I have a 3D solver for the incompressible Navier-Stokes equations which uses a FFT library for the Poisson equation with a uniform grid on all directions. In 2D the Poisson equation is given by: $$ ...
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2answers
280 views

Convergence problem for Poisson equation with periodic BC

I have written Poisson solvers using two different methods: A classic Jacobi scheme and one using the multigrid solver Hypre. I made up a couple of test cases ensuring the validity of those solvers. ...
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0answers
66 views

How to formulate Poisson's equation into flux eqution

I have a small 2D system I'm trying to model using a non-linear extension of Darcy's law for fluid flow in porous media. I'm primarily interested in the local flow velocity, not necessarily the ...
1
vote
2answers
298 views

Linear solvers: How to deal with a singular system? (Poisson equation with Neumann boundary conditions)

The question is in the context of iterative numerical solution of large PDE systems with Finite Differences or Finite Elements: Stating the Poisson equation with Neumann boundary conditions will lead ...
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2answers
312 views

Mixed formulation of the Poisson equation (FEM)

I'm solving the Dirichlet problem for the Poisson equation in a 2d domain $D$: $$ \begin{cases} \Delta u = 0 \quad \text{in $D$}, \\ u|_{\partial D} = u_0. \end{cases} $$ I'm interested in ...
5
votes
2answers
274 views

Solving Poisson equation with current BC using FEM

I have a 3-dimensional domain D, with 3 types of BC which I am trying to solve the Poisson equation on. $$-\nabla \cdot (\sigma(x,y,z)\nabla u)=0$$ Insulator (Neumann BC) Electrode set at some ...
4
votes
2answers
320 views

Solving Ax = b with sparse A and sparse b

Let's suppose I'm numerically solving the Poisson equation for a delta function source: $$ \nabla^2 f(x) = \delta(x-x') $$ I can represent the Laplacian $\nabla^2$ using the finite difference method ...
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0answers
118 views

Spectrum of the Laplace operator

I am studying the discretization of Poisson's equation in $1D$. In Matlab I created different discretization matrices (Laplace operator) according to different sizes of the mesh: ...
3
votes
1answer
1k views

Iteratively solving 3D Poisson equation in MATLAB

I have written a function that sets up a sparse matrix A and RHS b for the 3D Poisson equation in a relatively efficient way. The set-up is nothing fancy: I have extended the 2D 5-point stencil to an ...
1
vote
1answer
87 views

Surface Normal for 2D Finite Element Method

Consider this BVP with variable coefficients: $$-\nabla\cdot(a\nabla u) = f\ in\ \Omega$$ $$-n\cdot(a\nabla u) = \kappa(u-g_D)-g_N\ on\ \partial \Omega$$ where $a>0,f,\kappa>0,g_D,g_N$ are given ...
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votes
1answer
87 views

Solving Poisson equation while suffering from the curse of dimensionality

I have a heat transfer equation in a cube in $R^{100}$: $[0,1]\times[0,1]\times[0,1]\dots$: $$ \nabla^2 \varphi = f, $$ with boundary conditions set in a form that in the number of points $p_i$, ...
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vote
2answers
85 views

Finite difference equations representing semilinear elliptic PDE

I recently asked a question pertaining to the appliciation of Jacobi's method to a semilinear elliptic PDE (Poisson's equation) $$ \nabla^2u = -\rho~e^{-u} $$ A more efficient method like the Bi ...
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2answers
78 views

Explicitly including boundary points in a set of finite-difference equations

Consider the 1D poisson equation $$ \frac{d^2 u}{dx^2} = -\rho $$ with Dirichlet boundary conditions $u(0) = u(l) = g$. Using a finite difference scheme, with a 5-point grid $u_1,u_2,u_3,u_4,u_5$ (...
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1answer
240 views

Jacobi iteration for finite difference: when to stop?

I implemented a finite difference scheme to solve Poisson's equation in a 2D grid in C. I solve the system by using Jacobi iteration. Everything works fine until I use a while loop to check whether it ...
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votes
1answer
136 views

Use Finite Difference Discretization to find approximate solution to the Poisson's equation

I've just been introduced to the Poisson's equation. I've never had the need to dealt with PDE, so I'm a bit lost. Apparently we can compute an approximate solution of the Poisson's equation $$\frac{...
0
votes
1answer
834 views

Solving the Poisson equation with Neumann Boundary Conditions - Finite Difference, BiCGSTAB

I am trying to solve the Poisson equation in a rectangular domain using a finite difference scheme with a rectangular mesh. I have happily generated the matrix system of equations Ax = b which is ...
1
vote
1answer
107 views

Boundary treatment with higher order methods

I wrote a code which solves the 2D Poisson equation with homogeneous Dirichlet BC everywhere and a source term of -1. I am using the classical Jacobi iteration method. The grid is $N_x \, \mathrm{x} \,...
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vote
2answers
480 views

Boundary conditions in conjugate gradient method for poisson's equation

I want to use the conjugate gradient method to solve poisson's equation in an electrostatic setup: \begin{align} \rho=-\nabla^2\phi \end{align} I am however a little confused when it comes to the ...
0
votes
2answers
124 views

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,...
3
votes
1answer
258 views

Discretize Poisson equation with derivative of delta function as source

Consider the PDE \begin{equation} \frac{d^2}{dx^2} g(x) = \frac{d}{dx} \delta(x-x_0), \end{equation} with $x, x_0 \in [0,1]$ and $g(0)=g(1)=0$. What is the best method to discretise the derivative of ...
3
votes
1answer
207 views

How do I avoid divide-by-zero when solving the Poisson equation with Fourier transforms?

I wanted to try to implement part of the method in the following article using Fourier transforms. http://www.shodor.org/media/content/jocse/student_submissions/nocito2010/nocito2010_pdf Right now I ...
3
votes
0answers
145 views

Solving a nonlinear poisson equation via variational minimization

I am kind of new in finite elements and I am solving simple "Poisson nonlinear" problem. $- \nabla ((1 + u^2) \nabla u) = f$ $u = 0 \ \text{on} \ \Omega $ I am using Newton solver, where I have ...
2
votes
1answer
89 views

Boundary elements method — calculation of solid angle

I am developing a BEM code based on a deal.ii tutorial, see https://www.dealii.org/8.3.0/doxygen/deal.II/step_34.html . Consider the Poisson equation $$ \Delta u=-f\,, $$ and its Green's function $G\...
4
votes
0answers
129 views

Poisson equation in frequency domain

I need some help in numerically solving the nonlinear Poisson's equation for electrons in frequency domain. The steady-state equation is: \begin{equation} \nabla.(\epsilon\nabla\varphi) = q\left(n_i\...
6
votes
2answers
3k views

Poisson equation finite-difference with pure Neumann boundary conditions

I'm trying to solve a 1D Poisson equation with pure Neumann boundary conditions. I've found many discussions of this problem, e.g. 1) Poisson equation with Neumann boundary conditions 2) Writing the ...
2
votes
0answers
91 views

an ideal fluid in a reservoir

Let us imagine that we have an ideal fluid in a reservoir, not full, which means there is a free surface of an ideal fluid on top, this reservoir makes no movements. This problem mathematical ...
2
votes
1answer
68 views

Solving new linear system that comes from an $p$ enrichment

Let's say I am solving a simple Poisson problem using a Mixed (DG) finite element method. If we use orthogonal polynomials as basis functions we can write the finite-dimensional linear system as $$ ...
2
votes
1answer
215 views

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 ...
8
votes
1answer
297 views

Increasing V-cycles for constant Coarsest Grid Size and increasing Fine Grid size

Problem statement 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. ...
2
votes
2answers
1k views

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 ...
3
votes
2answers
242 views

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 ...
0
votes
1answer
131 views

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 ...
3
votes
1answer
508 views

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 ...
3
votes
0answers
340 views

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 ...
0
votes
1answer
37 views

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 ...
1
vote
1answer
1k views

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, ...
4
votes
2answers
2k views

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 ...
1
vote
0answers
52 views

Finding a scalar field in order to generate a solenoidal vector from a given vector

I am working on generating a (complex) solenoidal vector field $\mathbf{A}$ from a prescribed (complex) vector $\mathbf{a}$ and the gradient of a scalar, $b$, such that $$\mathbf{A} = \mathbf{a} + \...
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votes
1answer
660 views

explain the difference between 1D Poisson solvers

I compared 2 methods for 1 dimensional Poisson equation solution. One is finite-difference method, "Successive Overrelaxation" from http://www.cs.berkeley.edu/~demmel/cs267/lecture24/lecture24.html; ...
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vote
1answer
311 views

3D Poisson equation, Fourier and Chebyshev

I am currently trying to solve the 3D Poisson equation with a Chebyshev discretisation in the $z$ direction (from -1 to 1) and Fourier in the $x$ and $y$ (from $-\pi$ to $\pi$) I have taken the code ...
0
votes
1answer
162 views

jump conditions for Poisson/Darcy equation in primal form versus mixed form

Consider the Darcy equation, $$\mathbf{v} + \dfrac{k}{\mu_0}\nabla p = \mathbf{f} \\ \mathrm{div}\; \mathbf{v} = 0$$ If the coefficient $k$ is piecewise constant across an interface $\Gamma$ in the ...
4
votes
2answers
396 views

Choice of spaces for mixed formulation for Poisson Equation Or Darcy equation

Consider the mixed formulation of the Poisson/Darcy system for a region $\Omega$: $\alpha \mathbf{v} + \nabla p = f \\ \mathrm{div}[\mathbf{v}] = 0 $ with the boundary conditions $\mathbf{v}\cdot ...
3
votes
1answer
378 views

Jump condition for elliptic equation in standard finite element method

Consider the elliptic PDE $ -\mathrm{div}(k(x)\nabla u) = f(x) \in \Omega$ and $u = 0$ on $\partial \Omega$ If $k(x)$ is piecewise constant across an interface $\Gamma$ in the domain, then we ...
5
votes
3answers
2k views

Discrete Poisson Equation with Pure Neumann Boundary Conditions

I'm trying to implement the Helmholtz-Hodge Decomposition in 2D, which states that a vector field is composed by a rotational free component, a divergence free component and a harmonic component. ...
4
votes
1answer
3k views

Poisson equation with Neumann boundary conditions

I'm trying to solve the Poisson equation with pure Neumann boundary conditions, $$ \nabla^2\phi = \rho \quad in \quad \Omega\\ \mathbf{\nabla}\phi \cdot \mathbf{n} = 0 \quad on \quad \partial \Omega $$...
4
votes
1answer
336 views

Finite differences vs. elements: Accuracy and implementation

I am trying to solve the 2D Poisson equation numerically: $ \frac{\partial ^2 \phi}{\partial x^2} + \frac{\partial ^2 \phi}{\partial y^2} = 1 $ with the Dirichlet boundary condition $\phi = 0$. I ...