Energy Norms

📐 Definition

For a field $u$ on domain $\Omega$ ,

\|u\|_{E}^{2} = \int_{\Omega} |u(x)|^{2}\,dx

with discrete analog $\|u\|_{E}^{2} = \sum_{i} |u_i|^{2}\,\Delta x$ on uniform grids.

Domain and Codomain

Defined for square-integrable fields $u \in L^{2}(\Omega)$ (or discrete $\ell^{2}$ sequences). Output is nonnegative real; $\|u\|_{E} = 0$ iff $u=0$ almost everywhere.

⚙️ Key Properties

Quadratic, homogeneous, and satisfies the parallelogram law. For complex fields, $|u|^{2}$ uses modulus squared and corresponds to physical energy densities in many models.

On $2\pi$ -periodic domains with Fourier series coefficients

c_k = \frac{1}{2\pi}\int_{-\pi}^{\pi} u(x)\,e^{-ikx}\,dx,

Parseval’s identity gives

\int_{-\pi}^{\pi} |u(x)|^2\,dx = 2\pi \sum_{k\in\mathbb{Z}} |c_k|^2

(up to the chosen Fourier normalization). Normalizing by volume yields RMS magnitude.

🎯 Special Cases and Limits

Parseval identity links physical-space energy to spectral energy on periodic domains.
Volume-normalized energy corresponds to RMS $^2$ .

RMS is the volume-normalized energy norm; complex magnitude is the pointwise quantity integrated to form energy.

Usage in Oakfield

Oakfield uses energy-like norms primarily via per-field diagnostics and the thermostat operator:

Diagnostics compute RMS and related amplitude statistics; for complex fields this is based on $|u|^2 = \Re(u)^2 + \Im(u)^2$ .
Thermostat regulation targets the mean energy $E = \mathrm{mean}(|u|^2)$ and uses it as feedback to regulate dynamics.
Spectral operators (e.g. linear_dissipative) explicitly damp modes, and energy/norm summaries help verify the effect.

Historical Foundations

📜 L2 Energy and Parseval

Energy norms are closely tied to the $L^2$ norm in analysis and to Parseval/Plancherel identities in Fourier theory.

🌍 Modern Perspective

They are standard diagnostics for stability and conservation in numerical PDE solvers.

📚 References

Evans, Partial Differential Equations
Stein & Shakarchi, Fourier Analysis