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

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2
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2answers
155 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
139 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
75 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
100 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 ...
2
votes
0answers
112 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
25 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
433 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, ...
3
votes
2answers
430 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
44 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} + ...
-2
votes
1answer
204 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; ...
1
vote
1answer
123 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
97 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
137 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 ...
2
votes
1answer
133 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 ...
4
votes
3answers
651 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. ...
2
votes
1answer
706 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
284 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 ...
1
vote
1answer
131 views

Numerically solving the polar poisson equation [closed]

I want to solve the Poisson equation for a 2D polar system: $$\Delta_r f(r) = u(r)$$ with the Laplace operator: $ \Delta_r f(r) = \frac{1}{r}\frac{\partial}{\partial r} \left[r ...
1
vote
0answers
161 views

finite volume for diffusion equation with anisotropic (tensor) coefficient

Consider the scalar PDE for $u$ with Dirichlet boundary conditions: $\mathrm{div}(\mathcal{K}\nabla u) = f\; \forall x\; \in \Omega \subset R^2$, $u = 0 \; \forall \; x\;\in \partial\Omega$ ...
3
votes
1answer
264 views

Recommendations on FEM software for implementing Nitsche's method on interfaces between matching meshes?

Suppose: I have two domains, $\Omega_{1} = [0, 1/2] \times [0, 1]$ and $\Omega_{2} = [1/2, 1] \times [0, 1]$. The domains share an interface $\Gamma = \{1/2\} \times [0, 1] = \partial\Omega_{1} \cap ...
2
votes
2answers
177 views

data structures for efficient/easy implementation of finite volume method for 2D Poisson equation

My question is about implementation alone. Consider a square domain with regular square, cell centred finite volumes. This is for the multiscale finite volume method (Jenny and Lunati) I need to ...
1
vote
0answers
172 views

Assembling sparse matrix in PETSC for Poisson equation

I am a novice at PETSC, and I have been trying to write an FVM code for steady heat conduction in 2D using PETSC (square, regular grid, Dirichlet boundaries) Since the large matrix , say A, will be ...
3
votes
2answers
254 views

Poisson-Nernst-Planck equations with ill-conditioned sparse matrix

I am trying to solve Poisson-Nernst-Planck system of equations for ions diffusion problem using finite volume method. Nernst-Planck equation for mass transport and Poisson equation for electrostatic ...
7
votes
2answers
144 views

For a non-linear PDEs should the source term be discretised at $u_j$ or averaged over $(u_{j+1} + u_{j-1})/2$?

The non-linear Poisson equation in one-dimension, $$ 0 = \frac{\partial^2u}{\partial x^2} - f(u) $$ can be discretised as to give, $$ u_{j-1} -2u_{j} + u_{j+1} = h^2 f(u_j) $$ where $h$ is the ...
4
votes
1answer
82 views

Problem with Static 2D Heat Eqaution

So I am basically trying to solve the simplest case possible for the heat equation: on the domain [0;1] x [0,1] with boundary conditions 1.5 on one face of the square and 1.0 on all other faces. I ...
3
votes
0answers
280 views

assembly matrices in finite element method

I'm trying to construct the right–hand side of my 2D Poisson's equation in Matlab. I used the vertex rule in order to approximate the integral: ...
10
votes
1answer
994 views

When is Newton-Krylov not an appropriate solver?

Recently I have been comparing different non-linear solvers from scipy and was particularly impressed with the Newton-Krylov example in the Scipy Cookbook in which they solve a second order ...
5
votes
1answer
159 views

Is is possible to mix finite-volume and finite-difference discretisations when solving a coupled non-linear system?

In my last question I noticed a peculiar error when solving the Poisson equation using finite volume method (FVM), Peculiar error when solving the Poisson equation on a non-uniform mesh (1D only) ...
3
votes
2answers
253 views

Full Multigrid Performance for Poisson's equation using Higher Order Compact scheme as a Gauss Seidel smoother

I have a question regarding the FMG (Full Multi Grid) performance while computing Poisson's equation using Higher Order Compact discretization. I am using a sixth order compact scheme to discretize ...
6
votes
2answers
411 views

Peculiar error when solving the Poisson equation on a non-uniform mesh (1D only) finite volume method

I have been trying to debug this error the last few days I wondered if anybody has advice on how to proceed. I am solving the Poisson equation for a step charge distribution (a common problem in ...
3
votes
1answer
126 views

boundary oscillations with Robin boundary conditions

When solving Poisson's equation on the unit square $\Omega$ with homogeneous Dirichlet boundary conditions for $x=0$ and Robin-type conditions at the rest of the boundary, $$ \begin{cases} -\Delta u = ...
7
votes
3answers
1k views

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 ...
11
votes
3answers
242 views

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 ...
2
votes
2answers
618 views

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, ...
14
votes
1answer
781 views

Convergence rate of FFT Poisson solver

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 + ...
4
votes
1answer
206 views

Mehrstellenverfahren for Poisson?

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 ...
2
votes
1answer
378 views

Prolongation/Restriction Operator in Multigrid

In Multigrid, using Poisson's equation, does the equality below always hold regardless of what type of boundary conditions you use? $$ R= c\cdot I^T, \text{ for some constant }c $$ where $R$ and $I$ ...
4
votes
2answers
1k views

Testing 1D Poisson Solver

I'm trying to test a simple 1D Poisson solver to show that a finite difference method converges with $\mathcal{O}(h^2)$ and that using a deferred correction for the input function yields a convergence ...
10
votes
2answers
5k views

Writing the Poisson equation finite-difference matrix with Neumann boundary conditions

I am interested in solving the Poisson equation using the finite-difference approach. I would like to better understand how to write the matrix equation with Neumann boundary conditions. Would someone ...
3
votes
1answer
1k views

Finite Element Method: 2-D Poisson's Equation in Matlab, Gaussian quadrature

I'm having trouble understanding how to code 2-D Poisson's Equation with Dirichlet boundary conditions. What I have thus far is Constructed square mesh with triangular elements Assembled stiffness ...
5
votes
1answer
270 views

Poisson solver on unstructured mesh

For the 2D Poisson equation, there exist on finite difference mesh, some code taking $O(n \log(n))$ operations to solve it on a mesh with $n$ nodes. They rely on Fast Fourier Transform or Block Cyclic ...
2
votes
2answers
422 views

Solving Poisson equation with free boundaries and adaptively refined mesh

Assume we want to solve the Poisson equation $$ \Delta u = f $$ with free (Neumann) boundary conditions. So, the right hand side function $f$ must fulfill the compatibility condition to integrate to ...
5
votes
1answer
180 views

How far is a non-symmetric discretization of an elliptic operator from the continuous operator itself?

I am investigating the accuracy and stability properties of a non-symmetric discretization of a Poisson problem. The problem originates from a ghost fluid discretization of the projection step of a ...
3
votes
1answer
242 views

How to solve a problem with structure similar to a finite difference discretization of the 2D Poisson equation, but with non-symetric coefficients?

Recently, I've been asking about methods to solve a finite difference discretization of the 2D Poisson equation (see here and here) of the form: $$U_{i-1,j} + U_{i+1,j} -4U_{i,j} + U_{i,j-1} + ...
9
votes
1answer
523 views

Which fourier series is needed to solve a 2D poisson problem with mixed boundary conditions using Fast Fourier Transform?

I have heard that a fast fourier transform can be used to solve the poisson problem when the boundary conditions are all one type... Sine series for dirichlet, cosine for neumann, and both for ...