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One-dimensional SIPDG jump and penalty calculation

Example

This is a self-contained one-dimensional interval specialization of the jump convention in Jump and average conventions for symmetric interior-penalty DG. Let $K^{-} = (0, 1 / 2)$ and $K^{+} = (1 / 2, 1)$ , take unit coefficient, and set $h_{F} = 1 / 2$ at the interior point face. Define $v (x) = x$ on $K^{-}$ and $v (x) = x - 1$ on $K^{+}$ . Then $v (0) = v (1) = 0$ , while the interior jump at $x = 1 / 2$ is $1$ and the average derivative is $1$ . The face contributions are $- {v^{'}} [v] - {v^{'}} [v] + γ h_{F}^{- 1} [v]^{2}$ . With penalty parameter $γ = 1$ , the bilinear form has value $a_{h} (v, v) = 1$ .

Facts & Assumptions

Given: The two-cell interval partition, unit coefficient, counting measure on all point faces, $h_{F} = 1 / 2$ at each face (the common interval-cell length), the piecewise affine function $v$ , and $γ = 1$ . For functions $v, w$ on these cells, define $a_{h} (v, w) : = \sum_{K \in {K^{-}, K^{+}}} \int_{K} v^{'} w^{'} d x - \sum_{F \in {0, 1 / 2, 1}} (q_{F} (v) [w]_{F} + q_{F} (w) [v]_{F}) + γ \sum_{F \in {0, 1 / 2, 1}} h_{F}^{- 1} [v]_{F} [w]_{F},$ where $q_{1 / 2} (v) = {v^{'}}_{1 / 2}$ , $q_{0} (v) = - v^{'} (0 +)$ , and $q_{1} (v) = v^{'} (1 -)$ ; the endpoint jumps are the one-sided value traces and all face integrals use counting measure.

[F1]

On an interior face the scalar jump is the minus trace minus the plus trace, and the average is their arithmetic mean. (Jump and average conventions for symmetric interior-penalty DG)

Verification

1.1givenF1algebra

The derivative is $1$ on both cells, so the broken volume term is $\int_{0}^{1 / 2} 1 d x + \int_{1 / 2}^{1} 1 d x = 1$ . At the interior point $x = 1 / 2$ , [F1] gives $[v] = 1 / 2 - (- 1 / 2) = 1$ and ${v^{'}} = 1$ , while both boundary traces vanish.

2.1givenstep 1.1algebra∎

The interior consistency terms contribute $- 2 {v^{'}} [v] = - 2$ , and the penalty contributes $γ h_{F}^{- 1} [v]^{2} = 1 \cdot 2 \cdot 1 = 2$ . Adding these to the volume term from step 1.1 gives $a_{h} (v, v) = 1 - 2 + 2 = 1$ .

Depends on

Jump and average conventions for symmetric interior-penalty DG

Used by

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