Stationary Allen-Cahn FEM
This example solves a nonlinear stationary Allen-Cahn problem using classical FEM machinery.
Problem Setup
The weak form corresponds to a stationary Allen-Cahn equation with a nonlinear cubic term.
Step 1: Define the Nonlinear Weak Residual
The script expresses the nonlinear form directly in terms of FEM operators rather than pointwise PINN losses.
Step 2: Build the Jacobian and Nonlinear Solve Loop
Because the problem is nonlinear, a residual alone is not enough; the script also uses Jacobian information for iterative solution.
Step 3: Solve With a Classical Nonlinear Method
A SciPy-style root or nonlinear solve routine is used to converge the weak-form system.
What To Notice
- Weak-form nonlinear solves are structurally different from PINN optimization.
- This example is useful for comparing classical and neural treatments of the same PDE family.
- It also shows how jNO's weak-form abstractions extend beyond linear problems.