Struct SobolevSpace 

pub struct SobolevSpace {
    pub n: usize,
    pub dx: f64,
    pub k: usize,
}

A discrete Sobolev space H^k on a uniform grid.

Provides H^k norms, weak derivative approximation (finite differences), and trace operator evaluation.

Fields

n: usize

Number of grid points.

dx: f64

Grid spacing.

k: usize

Sobolev order k.

Implementations

impl SobolevSpace

pub fn new(n: usize, dx: f64, k: usize) -> Self

Create a new SobolevSpace with n points, spacing dx, and order k.

pub fn norm(&self, f: &[f64]) -> f64

Compute the H^k norm using finite-difference weak derivatives.

‖f‖²_{H^k} = Σ_{j=0}^{k} ‖D^j f‖²_{L²}

pub fn weak_derivative(f: &[f64], dx: f64) -> Vec<f64>

Compute the first-order weak derivative using central differences.

Forward/backward differences are used at the boundary.

pub fn trace(&self, f: &[f64]) -> [f64; 2]

Evaluate the trace operator: return the boundary values \[f(0), f(n-1)\].

pub fn seminorm(&self, f: &[f64]) -> f64

Compute the seminorm |f|_{H^k} = ‖D^k f‖_{L²}.

pub fn is_in_space(&self, f: &[f64]) -> bool

Check whether f lies in H^k (finite H^k norm).

pub fn inner_product(&self, f: &[f64], g: &[f64]) -> f64

Compute the H^k inner product ⟨f, g⟩_{H^k} = Σ_{j=0}^{k} ⟨D^j f, D^j g⟩.

Trait Implementations

impl Clone for SobolevSpace

fn clone(&self) -> SobolevSpace

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for SobolevSpace

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.