Finite Element Method for Poisson Problems

A Python implementation of the Finite Element Method (FEM) to solve the 2D Poisson equation, based on the Bachelor Thesis.

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📌 Overview

This project provides a full FEM pipeline:

• Triangular mesh generation
• Assembly of stiffness matrix & load vector
• Dirichlet and Neumann boundary conditions
• Solution via Conjugate Gradient (CG)
• Optional Incomplete Cholesky preconditioning
• Visualization of the solution

Supports linear, quadratic, and cubic elements.

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🧠 Problem

Solve the Poisson equation:

-Δu = f  in Ω

with mixed boundary conditions:

• Dirichlet: u = g_D
• Neumann: ∇u · n = g_N

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⚙️ Installation (Poetry)

poetry install
poetry shell

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▶️ Usage

Run an example:

poetry run python room_temperature.py

or

poetry run python electric_field_ion_channels.py

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🧩 Project Structure

• assemble_system.py — global FEM assembly
• auxiliary_assemblage_functions.py — element matrices & integration
• cg.py — Conjugate Gradient solver
• cholprec.py — incomplete Cholesky preconditioner
• preprocessing.py — mesh handling
• postprocessing.py — visualization

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🌡 Example: Room Temperature Simulation

Simulates heat diffusion with:

• internal heat source
• fixed boundary temperature
• realistic domain geometry

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🔧 Features

• Multiple element orders (P1, P2, P3)
• Sparse matrix assembly
• Krylov solver (CG)
• Preconditioning support
• Gaussian quadrature integration

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