Fluid model

Poisson

Weak discretization of Poisson's equation with a diffusion matrix, stabilization and an optional time-dependent right-hand side.

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Overview

Poisson is the standalone elliptic field-solve model used for weak

diffusion/Poisson problems. It is also the building block reused by other models for initial electrostatic solves.

Use cases

This model is appropriate for:

elliptic benchmark problems
electrostatic field solves with prescribed source terms
testing weak Poisson discretizations and boundary handling

Governing equations

PDEs solved by model:

Find $ϕ \in H^{1}$ such that

- \nabla \cdot D_{0} (x) \nabla ϕ + n_{0} (x) ϕ = ρ (t, x)

where $n_{0}, ρ (t) : Ω \to R$ are real-valued functions, $ρ (t)$ is parametrized by time $t$ , and $D_{0} : Ω \to R^{3 \times 3}$ is a positive matrix. Boundary terms from integration by parts are assumed to vanish.

Normalization

The coefficient scaling is

D^= n^/ x^^{2}, ρ^= n^.

No dedicated velocity normalization is used.

Discretization

Time integration is performed by the following propagators (in sequence):

TimeDependentSource (if with_t_dep_source is True)
PoissonSolve

Diagnostics

The following scalars are tracked during simulation:

No default scalar diagnostics are defined by this model.

Examples

Poisson convergence on a distorted mesh

A cosine source on a periodic box, solved with splines of degree 1 to 3 on a mesh that the Colella mapping has bent. The error of the potential against the exact solution falls with the mesh width h at least as fast as h^(p+1) for a spline degree p, on the curved mesh as well as on the straight one. The prescribed source and reference potential are $\rho(x,y)=\cos(\pi x)\cos(2\pi y/3),\qquad \phi_{\mathrm{exact}}=\frac{\rho(x,y)}{\pi^2+(2\pi/3)^2},$ on $[0,2)\times[0,3)$ .

Poisson time-dependent source

Drive a 1D Poisson solve with an oscillating cosine-mode charge density and compare Struphy’s FEEC potential against the exact solution, at every time step. The prescribed source is $\rho(x,t)=0.1\cos(2\pi x/5)\cos(2\pi t),$ on the periodic interval $-5\le x<5$ .