At this week's livestream ( Differential Geometry and the Metric Identities ), we're discussing how to handle complex geometry using unstructured meshes and isoparametric elements. When mapping isoparametric elements from physical to computational space, we need to be careful when calculating the Jacobian contravariant basis vectors so that we ensure satisfaction of the metric identities.

Lack of satisfaction of the metric identities can lead to spurious flows that also prohibit proofs of entropy conservation. Although the incurred error is spectrally accurate, careful formulation of the metric terms is necessary to eliminate spurious flows and as part of the proof for developing an entropy stable method.

These notes cover the derivation of how to compute derivative operations under coordinate transformation and they summarize the development of the conservative and curl invariant forms of the Jacobian weighted contravariant basis vectors that both satisfy the discrete metric identities.

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