Helmholtz Sturm-Liouville Problem

Helmholtz Sturm-Liouville Problem#

This example solves the Helmholtz differential equation, which is a classic Sturm-Liouville problem:

\[ u'' + u = 0, \quad 0 < x < 3 \]

or $\( u'' + \mu^2 u = 0, \quad 0 < x < 1 \)$

with boundary conditions: $\( u(0) = 0, \quad u(3) = \sin(3) \)\( or \)\( u'(0) = 0, \quad u(1) + u'(1) = \cos(\mu) - \mu \sin(\mu) \)$

The exact solution to this problem is \(\sin(x)\) or \(\cos(\mu x)\).

Mathematical Background#

Helmholtz’s differential equation is a special case of the Sturm-Liouville problem, which has the general form:

\[\frac{d}{dx}\left(p(x)\frac{du}{dx}\right) + q(x)u + \lambda r(x)u = 0\]

For Helmholtz’s equation, we have:

  • \(p(x) = 1\)

  • \(q(x) = 0\)

  • \(r(x) = 1\)

  • \(\lambda = \mu^2\)

Discretization#

The equation is discretized using mimetic finite difference operators. The spatial derivative operators are constructed with a specified order of accuracy \(k\).

The discrete system is:

\[A u = b\]

where:

  • \(A = L + I\)

  • \(L\) is the mimetic Laplacian

  • \(I\) is the identity matrix

Boundary conditions are applied using RobinBC or MixedBC.

These examples are implemented in:

Results#

The numerical solutions closely match the exact solutions, which are \(\sin(x)\) or \(\cos(\mu x)\).