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One-dimensional discrete summation by parts with boundary terms

Statement

Let $N \geq 2$ , $h > 0$ , and let $u = (u_{0}, \dots, u_{N})$ and $v = (v_{0}, \dots, v_{N})$ be complex grid functions. With the finite-interval inner products and differences of Uniform periodic, infinite, and Dirichlet grid conventions, $⟨ u, - Δ_{h} v ⟩_{h} = ⟨ D_{+} u, D_{+} v ⟩_{e, h} + u_{0} \overline{(D_{+} v)_{0}} - u_{N} \overline{(D_{-} v)_{N}} .$ In particular, if $u_{0} = u_{N} = 0$ , the two boundary terms vanish. On the subspace of grid functions with zero endpoints, $- Δ_{h}$ is self-adjoint and nonnegative in the interior grid inner product, with $⟨ u, - Δ_{h} u ⟩_{h} = ∥ D_{+} u ∥_{e, h}^{2}$ .

Facts & Assumptions

Given: $N \geq 2$ , $h > 0$ , and complex grid values $u_{j}, v_{j}$ on $j = 0, \dots, N$ .

[F1]

The interior and edge inner products and differences are defined in Uniform periodic, infinite, and Dirichlet grid conventions.

Proof

1.1F1algebra

Put $d_{j} = v_{j} - v_{j - 1}$ for $j = 1, \dots, N$ . Expanding the second difference gives $⟨ u, - Δ_{h} v ⟩_{h} = - \frac{1}{h} \sum_{j = 1}^{N - 1} u_{j} \overline{(d_{j + 1} - d_{j})} .$ Shift the index in the $d_{j + 1}$ sum and telescope: $- \sum_{j = 1}^{N - 1} u_{j} \overline{(d_{j + 1} - d_{j})} = \sum_{j = 1}^{N} (u_{j} - u_{j - 1}) \overline{d_{j}} + u_{0} \overline{d_{1}} - u_{N} \overline{d_{N}} .$ Dividing by $h$ identifies the edge sum and boundary terms in the displayed identity.

2.1step 1.1F1∎

When $u_{0} = u_{N} = 0$ , step 1.1 reduces to $⟨ u, - Δ_{h} v ⟩_{h} = ⟨ D_{+} u, D_{+} v ⟩_{e, h}$ . Applying this identity after exchanging $u, v$ and taking complex conjugates gives $⟨ - Δ_{h} u, v ⟩_{h} = ⟨ u, - Δ_{h} v ⟩_{h}$ , so $- Δ_{h}$ is self-adjoint on the zero-endpoint subspace. Setting $u = v$ gives $⟨ u, - Δ_{h} u ⟩_{h} = ∥ D_{+} u ∥_{e, h}^{2} \geq 0$ .

Depends on

Uniform periodic, infinite, and Dirichlet grid conventions

Used by

Coercive discrete energy for the Dirichlet leapfrog wave scheme Theorem

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Endre Süli, B6.1 Numerical Solution of Partial Differential Equations (standard reference, not scraped)