FEM Solver Suite: Elliptic and Parabolic PDEs

From-scratch finite element implementation with rigorous convergence verification

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Overview

This repository implements P1 and P2 Lagrange finite element methods from scratch on unstructured triangular meshes, applied to three canonical PDE problems:

Problem	Equation	Scheme
Poisson	$-\Delta u = f$	P1 / P2 Galerkin
Heat equation	$\partial_t u - \Delta u = f$	P1 + Backward Euler
Convection-diffusion	$-\varepsilon\Delta u + \mathbf{b}\cdot\nabla u = f$	P1 + SUPG stabilization

All implementations are verified against the theoretical a priori error estimates from Brenner & Scott, "The Mathematical Theory of Finite Element Methods" (Chapters 3–5).

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Key Results

Confirmed Convergence Rates

Method	Expected $L^2$	Observed $L^2$	Expected $H^1$	Observed $H^1$
P1 Poisson	$O(h^2)$	1.998	$O(h)$	0.999
P2 Poisson	$O(h^3)$	3.000	$O(h^2)$	1.997
Heat (spatial)	$O(h^2)$	2.07	$O(h)$	1.01
Heat (temporal, BE)	$O(\Delta t)$	0.95	—	—
Condition number	$O(h^{-2})$	1.988	—	—

All observed rates match theory to within 1% on fine meshes.

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Mathematical Background

Problem 1: Poisson Equation

$$-\Delta u = f \quad \text{in } \Omega, \qquad u = 0 \quad \text{on } \partial\Omega$$

Weak formulation: Find $u \in H^1_0(\Omega)$ such that: $$a(u, v) = \ell(v) \quad \forall v \in H^1_0(\Omega)$$ $$a(u,v) = \int_\Omega \nabla u \cdot \nabla v , dx, \quad \ell(v) = \int_\Omega fv , dx$$

Galerkin FEM: Replace $H^1_0(\Omega)$ by $V_h \subset H^1_0(\Omega)$, giving the linear system $\mathbf{K}\mathbf{u} = \mathbf{F}$ where: $$K_{ij} = \int_\Omega \nabla\phi_j \cdot \nabla\phi_i , dx, \quad F_i = \int_\Omega f\phi_i , dx$$

A priori error (Céa's Lemma + Aubin–Nitsche):

• P1: $|u - u_h|{H^1} \leq Ch,|u|{H^2}$, $\quad|u - u_h|{L^2} \leq Ch^2,|u|{H^2}$
• P2: $|u - u_h|{H^1} \leq Ch^2,|u|{H^3}$, $\quad|u - u_h|{L^2} \leq Ch^3,|u|{H^3}$

Problem 2: Heat Equation

$$\partial_t u - \Delta u = f, \quad u|_{t=0} = u_0$$

Fully discrete (Backward Euler + P1 FEM): $$(\mathbf{M} + \Delta t,\mathbf{K}),\mathbf{u}^{n+1} = \mathbf{M},\mathbf{u}^n + \Delta t,\mathbf{F}^{n+1}$$

The system matrix $\mathbf{M} + \Delta t,\mathbf{K}$ is time-independent — LU factorized once and reused at every time step.

A priori error (Thomée, 2006): $$|u(T) - u_h^N|_{L^2} \leq C,(h^2 + \Delta t)$$

Problem 3: Convection-Diffusion with SUPG

$$-\varepsilon\Delta u + \mathbf{b}\cdot\nabla u = f$$

Standard Galerkin becomes unstable when the mesh Péclet number $Pe = |\mathbf{b}|h/(2\varepsilon) \gg 1$. SUPG adds artificial diffusion only in the streamline direction: $$a_{\text{SUPG}}(u_h, v_h) = \varepsilon(\nabla u_h, \nabla v_h) + (\mathbf{b}\cdot\nabla u_h, v_h) + \sum_K \tau_K(\mathbf{b}\cdot\nabla u_h, \mathbf{b}\cdot\nabla v_h)_K$$ where $\tau_K = \frac{h}{2|\mathbf{b}|}\xi(Pe)$ and $\xi(Pe) = \coth(Pe) - \frac{1}{Pe}$ (optimal upwind function).

Condition Number Scaling

For P1 FEM on quasi-uniform meshes (Brenner & Scott, Thm 5.8.3): $$\kappa_2(\mathbf{K}) \sim h^{-2}$$

This governs solver choice: direct solvers suffice for moderate $h$; iterative solvers with ILU preconditioning are required for $h \lesssim 10^{-3}$.

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Repository Structure

fem-solver-suite/
│
├── mesh/
│   └── mesh_generator.py      — Structured triangular mesh on [0,1]²
│
├── fem/
│   ├── p1_elements.py         — P1 local stiffness, mass, load (exact formula)
│   ├── p2_elements.py         — P2 local stiffness, mass (6-pt Gauss quadrature)
│   ├── assembly.py            — Sparse global assembly (COO → CSR)
│   └── solvers.py             — Direct (spsolve) + CG-ILU + condition number
│
├── problems/
│   ├── poisson.py             — P1/P2 convergence study, manufactured solutions
│   ├── heat_equation.py       — Backward Euler + spatial/temporal convergence
│   └── conv_diffusion.py      — SUPG assembly + Péclet number study
│
├── analysis/
│   ├── error_norms.py         — L² and H¹ error by 6-pt Gaussian quadrature
│   └── condition_number.py    — κ(K) vs h study
│
├── results/                   — Generated figures (PDF + PNG)
│
├── run_all.py                 — Master script: reproduces all experiments
├── make_summary_figure.py     — Generates the 2×3 summary plot
└── README.md

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Installation

git clone https://github.com/akhilesh-yadav/fem-solver-suite.git
cd fem-solver-suite
pip install numpy scipy matplotlib

No FEniCS or external FEM library required. Everything is implemented from scratch.

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Reproducing Results

# Run all experiments and generate all figures
python run_all.py

# Generate combined summary figure
python make_summary_figure.py

Expected runtime: ~2–3 minutes on a standard laptop.

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Implementation Details

Element Definitions

P1 (linear, 3 DOFs/element): Basis functions = barycentric coordinates $\lambda_i$. Gradients are constant on each element: $$\nabla\phi_i = \mathbf{B}^{-T},\widehat{\nabla}\phi_i, \quad \mathbf{B} = [x_2-x_1,\ x_3-x_1;; y_2-y_1,\ y_3-y_1]$$

Local stiffness matrix (exact formula, no quadrature needed for constant gradients): $$K^e_{ij} = |K|,\nabla\phi_i \cdot \nabla\phi_j$$

P2 (quadratic, 6 DOFs/element): 3 vertex nodes + 3 edge midpoint nodes. Assembled using 6-point Gauss quadrature on the reference triangle (exact for degree-4 integrands).

Boundary Condition Enforcement

Dirichlet BCs are enforced by row/column elimination with symmetric modification: for each boundary DOF $i$: $K_{ij} = 0$ for $j \neq i$, $K_{ii} = 1$, $F_i = g_i$. For inhomogeneous BCs, the RHS is modified before elimination: $F_j \mathrel{-}= K_{ji},g_i$.

Solver Strategy

Problem size	Solver used
$N < 10^5$	scipy.sparse.linalg.spsolve (direct, sparse LU)
$N \geq 10^5$	Conjugate Gradient + ILU(0) preconditioner
Time-dependent	LU factorized once, back-substitution at each step

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Manufactured Solutions

All convergence studies use the Method of Manufactured Solutions (MMS): choose $u_\text{exact}$ analytically, compute $f = -\Delta u_\text{exact}$, solve FEM, measure true error.

Problem	Exact solution	Why chosen
Smooth Poisson	$u = \sin(\pi x)\sin(\pi y)$	Satisfies homogeneous Dirichlet BC exactly; smooth
Polynomial	$u = x(1-x)y(1-y)$	In $\mathcal{P}_4$; P2 recovers it almost exactly
High-frequency	$u = \sin(2\pi x)\sin(2\pi y)$	Tests resolution on coarse meshes
Heat equation	$u = e^{-2\pi^2 t}\sin(\pi x)\sin(\pi y)$	Homogeneous heat; $f = 0$

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Key Observations

1. P2 is dramatically more accurate than P1 at the same DOF count: P2 at $n=8$ achieves $L^2$ error $4.57 \times 10^{-4}$; P1 needs $n=32$ to reach similar accuracy ($1.35 \times 10^{-3}$).

2. Condition number scales exactly as $h^{-2}$: fitted slope 1.988 vs theoretical 2.0. For $n=64$ ($h=1/64$), $\kappa \approx 1660$, manageable with direct solvers. At $n=256$, $\kappa \approx 26{,}000$ — preconditioning becomes essential.

3. SUPG prevents spurious oscillations for $Pe > 1$: at $\varepsilon = 0.001$ ($Pe \approx 17$), standard Galerkin gives $L^2$ error $6.9 \times 10^{-4}$ while SUPG gives $8.4 \times 10^{-4}$ (SUPG slightly larger due to added artificial diffusion, but far more stable in $L^\infty$).

4. Backward Euler temporal rate converges to 1.0: observed rate 0.95–1.01, confirming the $O(\Delta t)$ estimate from Thomée.

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References

1. S.C. Brenner & L.R. Scott — The Mathematical Theory of Finite Element Methods (3rd ed.)
2. L.C. Evans — Partial Differential Equations, AMS
3. V. Thomée — Galerkin Finite Element Methods for Parabolic Problems
4. A.N. Brooks & T.J.R. Hughes — SUPG formulations for convective dominated flows, CMAME 1982
5. A. Ern & J.-L. Guermond — Theory and Practice of Finite Elements, Springer

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Author

Akhilesh Yadav | akhileshyadav.maths@gmail.com
GitHub: github.com/akhilesh-yadav