Questions tagged [poisson-equation]
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8
questions
3
votes
1
answer
180
views
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 ...
- 31
0
votes
1
answer
85
views
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) (...
- 53
1
vote
0
answers
125
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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 ...
- 119
5
votes
0
answers
90
views
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 ...
- 51
0
votes
2
answers
138
views
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 (...
- 53
0
votes
1
answer
164
views
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, ...
- 53
1
vote
1
answer
85
views
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 ...
- 123
2
votes
0
answers
725
views
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 ...
- 570