Questions tagged [poisson-equation]

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My toy Laplace equation solver using finite-difference is unstable and I'm not sure why

I am trying to solve the variable-coefficient Laplace equation $$\partial(\epsilon\partial u) = (\partial\epsilon)(\partial u) + \epsilon\partial^2 u = 0$$using a finite difference scheme: $$\left(\...
DJames's user avatar
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3 votes
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Finite difference problem

I have a problem to resolve with the Finite Difference method in $[a,b]$: $$-\frac{d}{dx}(\alpha(x)\frac{du}{dx})= g(x),$$ with $\alpha(x) \in L^{\infty}$ continuous in $]a,c[$ and $]c,b[$ and ...
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Discretization of generalized kinetic term in 2D Poisson partial differential equation

A typical 2D Poisson PDE is given as $$\nabla^2\varphi(x, y)=f(x, y)$$ where the Laplacian term, $\nabla^2\varphi$, can to some degree be interpreted as the kinetic energy (given proper scaling) (...
Akhaim's user avatar
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Solving 2D Poisson equation with mixed boundary conditions in Python

I am trying to numerically solve the Poisson's equation $$ u_{xx} + u_{yy} = - \cos(x) \quad \text{if} - \pi/2 \leq x \leq \pi/2 \quad \text{0 otherwise} $$ The domain is the rectangle with vertices ...
user82261's user avatar
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How to get an "optimal point" for refinement in FEM adaptive mesh refinement?

Consider the following 1D problem \begin{align*} \begin{cases} \displaystyle -\frac{d^2u}{dx^2} = f(x), \hspace{0.5cm} x\in (a,b) \\[4mm] u(a) = u_{a}, \ \ u(b) = u_{b} \end{cases} \end{align*} I ...
Warren's user avatar
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Practical implementation of the discrete compatibility condtion

I've recently started looking into writing a finite-differences-based solver for the Poisson equation of the form $$\nabla\left(\varepsilon\nabla\varphi\right)=\rho$$ in 2D for arbitrary geometries (...
Akhaim's user avatar
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Can successive over-relaxation (SOR) method deal with ill-posed PDE BVP?

Recently, I've been struggling to understand the limitations and capabilities of the successive over-relaxation (SOR) method for boundary value problems which are ill-posed, such as, for instance, ...
Akhaim's user avatar
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How can I define an equipotential surface/volume in FEniCS?

I want to solve electrostatic problem for potential. Charge density and medium permittivity are known, so is the potential of a grounded surface. I know how I can implement that. But I would like to ...
abukaj's user avatar
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2 votes
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How to write a simple finite element solver in python in order to solve Poisson equation in 2D

I would like to write a simple finite element solver in python in order to solve 2D Poisson equation and then visualize it. $$ -\nabla^{2} u(x,y)=f(x,y), \quad x,y \quad in \quad \Omega\\ u(x,y) = u_D ...
Dude's user avatar
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