What is being solved physically?

The equation **−Δu=f** can model electrostatic potential from charge density, steady heat conduction with sources, or a membrane deflection under load. The interpretation changes, but the elliptic operator and boundary-condition ideas are the same.

Why are Neumann boundaries “natural” in FEM?

After integrating by parts, flux boundary terms appear on the right-hand side as **∫Γ_N g v ds**. Dirichlet values must be imposed by pinning degrees of freedom, while Neumann data enters directly as load contributions.

Is this a production-quality FEM code?

No. It is a teaching model on a structured triangular mesh with simple quadrature and a basic CG solver. Real FEM software adds mesh generation, higher-order elements, robust preconditioning, error estimators, and careful treatment of complex geometries.

Finite Element Poisson Solver (2D)

This page is a compact finite-element Poisson solver for −Δu = f on the unit square. The domain is split into a regular triangular mesh and each triangle uses linear P1 basis functions. For every element, the local stiffness matrix is assembled from the gradients of the basis functions, Kᵉ_ij = (b_i b_j + c_i c_j)/(4A), then accumulated into a sparse global system. Source terms are integrated by a centroid rule. In pure Dirichlet mode all boundary nodes are pinned to u=0, like a membrane clamped to a frame. In mixed mode, the left/right sides are Dirichlet while top/bottom edges contribute natural Neumann flux terms to the right-hand side. The resulting symmetric positive-definite system is solved by conjugate gradients. The left view colors the potential on the triangular mesh; the right view lifts the same scalar field into a “deformable membrane” surface.

Who it's for: Students in numerical PDEs, finite elements, electrostatics, and computational mechanics learning weak forms, triangular P1 elements, and boundary-condition handling.

Key terms

Finite element method
Poisson equation
Triangular mesh
P1 element
Weak form
Dirichlet boundary
Neumann boundary
Conjugate gradients

How it works

A compact 2D finite-element Poisson solver: assemble linear triangular P1 elements for -Δu=f, apply Dirichlet or natural Neumann boundaries, solve the sparse system by conjugate gradients, and view the potential as a colored membrane.

Key equations

Weak form: find u such that ∫Ω ∇u·∇v dΩ = ∫Ω f v dΩ + ∫Γ_N g v ds. P1 triangles give local stiffness K_ij^e = (b_i b_j + c_i c_j)/(4A); Dirichlet values are pinned.

Frequently asked questions

What is being solved physically?: The equation −Δu=f can model electrostatic potential from charge density, steady heat conduction with sources, or a membrane deflection under load. The interpretation changes, but the elliptic operator and boundary-condition ideas are the same.
Why are Neumann boundaries “natural” in FEM?: After integrating by parts, flux boundary terms appear on the right-hand side as ∫Γ_N g v ds. Dirichlet values must be imposed by pinning degrees of freedom, while Neumann data enters directly as load contributions.
Is this a production-quality FEM code?: No. It is a teaching model on a structured triangular mesh with simple quadrature and a basic CG solver. Real FEM software adds mesh generation, higher-order elements, robust preconditioning, error estimators, and careful treatment of complex geometries.

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Finite Element Poisson Solver (2D)

How it works

Key equations

Frequently asked questions

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Frequently asked questions