Legendre Sturm-Liouville Problem

Legendre Sturm-Liouville Problem#

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

\[ (1 - x^2) u'' - 2 x u' + n (n + 1) u = 0, \quad -1 < x < 1 \]

with Dirichlet boundary conditions: $\( u(-1) = -1, \quad u(1) = 1 \)$

The exact solutoin to this problem is the Legendre polynomial of degree \(n\), denoted as \(L_n(x)\). For \(n = 4\), the solution is \(L_4(x) = \frac{1}{8}(35x^4 -30x^2 + 3)\).

Mathematical Background#

Legendre’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 Legendre’s equation, we have:

  • \(p(x) = 1-x^2\)

  • \(q(x) = 0\)

  • \(r(x) = 1\)

  • \(\lambda = n(n+1)\)

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:

  • \(A = (1 - x^2) L - 2 x I_{FC} G + n(n+1) I\)

  • \(L\) is the mimetic Laplacian

  • \(G\) is the mimetic gradient

  • \(I_{FC}\) is the interpolation operator from faces to centers

  • \(I\) is the identity matrix

Boundary conditions are applied using RobinBC.

This example is implemented in:

Results#

The numerical solution closely matches the exact solution, which is the Legendre polynomial \(L_4(x) = \frac{1}{8}(35x^4 -30x^2 + 3)\).