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Boundary elimination returns exactly the finite Chebyshev collocation system

Statement

For the algorithm Boundary-eliminated Chebyshev collocation for a two-point Dirichlet BVP, a success output is the unique nodal vector with $U_{0} = g_{+}$ , $U_{N} = g_{-}$ satisfying

- (D_{N}^{(2)} U)_{i} + a (x_{i}) (D_{N} U)_{i} + b (x_{i}) U_{i} = f (x_{i}), 1 \leq i < N,

where $D_{N}$ is the exact Chebyshev–Gauss–Lobatto differentiation matrix and $D_{N}^{(2)} = D_{N} D_{N}$ . If the interior matrix is singular, the algorithm returns singular-interior-matrix and makes no claim about existence or uniqueness for the given right-hand side.

Facts & Assumptions

Given: An integer $N \geq 2$ , node values of $a, b, f$ , endpoints $g_{-}, g_{+}$ , the boundary-elimination algorithm, and its interior matrix $A$ .

[F1]

Boundary-eliminated Chebyshev collocation for a two-point Dirichlet BVP: $C_{i j} = - (D_{N}^{2})_{i j} + a (x_{i}) (D_{N})_{i j} + b (x_{i}) δ_{i j}$ , $A = (C_{i j})_{1 \leq i, j < N}$ , and $r_{i} = f (x_{i}) - C_{i 0} g_{+} - C_{i N} g_{-}$ .

[F2]

Chebyshev–Gauss–Lobatto differentiation is exact on the interpolant: for the interpolant $p$ through the full nodal vector $U$ , $D_{N} U = (p^{'} (x_{i}))_{i}$ ; applying $D_{N}$ again gives $(p^{''} (x_{i}))_{i}$ .

[F3]

Invertible matrix theorem: invertibility, full pivot rank, RREF $I$ , trivial nullspace and unique solvability are equivalent: an invertible square matrix has exactly one solution for every right-hand side.

Proof

1.1givenF1

On a success branch the algorithm sets $U_{0} = g_{+}$ and $U_{N} = g_{-}$ and solves $A U_{I} = r$ for the interior vector $U_{I} = (U_{1}, \dots, U_{N - 1})$ .

1.2givenF2

By [F2], $D_{N} U$ and $D_{N}^{(2)} U$ are respectively the first and second derivatives of the degree-at-most- $N$ nodal interpolant evaluated at the nodes.

2.1givenF1step 1.1step 1.2algebra

Expanding the $i$ th row of the full expression using [F1] gives $\sum_{j = 1}^{N - 1} C_{i j} U_{j} + C_{i 0} g_{+} + C_{i N} g_{-} = f (x_{i}),$ which is exactly the reduced equation $A U_{I} = r$ with the definition of $r_{i}$ . Combined with step 1.2, this is the stated collocation equation at every interior node.

2.2givenF1F3step 1.1algebra

Since the success branch has invertible $A$ , [F3] gives exactly one interior vector solving $A U_{I} = r$ . The prescribed endpoint values then give exactly one full nodal vector. The singular branch returns its declared status and asserts nothing about solvability of its right-hand side.

3.1step 1.1step 1.2step 2.1step 2.2∎

Steps 1.1–2.2 prove the success claim and the stated singular-matrix limit.

Depends on

Used by

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Sources

Jie Shen and Tao Tang, Spectral and High-Order Methods with Applications, §2.4 (standard reference, not scraped)
Lloyd N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, Ch. 8 (standard reference, not scraped)