This code provides a simple, pedagogical implementation of the finite element method for solving parabolic partial differential equations (PDEs) in one dimension.
This code is designed to illustrate a collection of fundamental methods/techniques for solving PDEs using the finite element method. It is recommended that a more high-performance computing platform such as FEniCS is used for real applications of the finite element method. Setting up and solving partial differential equations in FEniCS is extremely straight-forward and much more numerically stable than the more pedagogical code provided in this repository.
The finite element space basis functions which are implemented in this code are:
linear_basis- linear basis functionsquadratic_basis- quadratic basis functions
The desired set of basis functions can be specified by setting the basis variable in the model.py file.
There are also two choices of solver methods available:
Cholesky- Solve using a Cholesky factorization and triangular inversionConjugate Gradient- Solve using the conjugate gradient method
which can specified by the solve_method variable.
Moreover, there are two time-stepping schemes:
Backward Euler- The implicit backward Euler schemeCrank-Nicolson- The Crank-Nicolson / Theta=0.5 scheme
which can be specified by the time_stepping variable.
The code can be run by simply issuing the command:
user@host $ python model.py
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Finite Element Space: |
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- Spatial Domain = [0.00, 1.00]
- Time Interval = [0.00, 0.15]
- FEM Basis = Linear Basis Functions
- Time Stepping = Crank-Nicolson
- Solve Method = Conjugate Gradient
- Problem Dimension = 99
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FEM Solver Progress: |
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[ Forming Stiffness Matrix ]
[ Forming Mass Matrix ]
[ Computing Preconditioner ]
[ Solving Linear System ] (Step 101 of 101)
Solve Time: 44.92097 seconds
[ Saving Animation ]
This code will also generate and save an animation for the computed solution:
