Poisson 2D

anypinn create my-project --template poisson-2d

2D elliptic PDE forward problem. Solves for the solution field given a known source term and boundary conditions.

Background

The Poisson equation is one of the most fundamental elliptic PDEs in mathematical physics. It arises in electrostatics (computing the electric potential from a charge distribution), gravitational theory, and steady-state heat conduction. As a time-independent equation it describes equilibrium states, making it the simplest setting for demonstrating PDE residual and boundary condition constraints. The template uses a manufactured solution with an analytically known source term, so the forward solve can be verified exactly.

Governing Equations

\[ -\nabla^2 u = f(x, y) \]

with source term:

\[ f(x, y) = -2\pi^2 \sin(\pi x)\,\sin(\pi y) \]

where:

• \(u(x, y)\): solution field
• \(\nabla^2 = \dfrac{\partial^2}{\partial x^2} + \dfrac{\partial^2}{\partial y^2}\): 2D Laplacian
• \(f(x, y)\): prescribed source (code: source_fn)

Default Configuration

The generated template uses the following values.

Boundary conditions: \(u = 0\) on \(\partial\Omega\) (homogeneous Dirichlet on all four edges).

Domain: \((x, y) \in [0, 1]^2\)

In the inverse variant (--direction inverse), a scaling parameter \(k\) is introduced as \(-k\,\nabla^2 u = f\) with true value \(k = 1\) (code: TRUE_K).

Features Demonstrated

• PDEResidualConstraint for interior residual enforcement
• DirichletBCConstraint for boundary condition enforcement
• Forward problem (no parameter recovery)

Results