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Discrete Differential Forms for Numerical Relativity

# Discrete Differential Forms for Numerical Relativity -...

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Discrete Differential Forms for Numerical Relativity Ari Stern Applied & Computational Mathematics California Institute of Technology [email protected] 24th PCGM March 22, 2008

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Stern, Discrete Differential Forms for Numerical Relativity 24th PCGM, 3/22/08 Discrete Differential Forms To specify an n -dimensional mesh, we are given the following topological data for all k = 0 , 1 , . . . , n : A list of oriented k -cells (vertices, edges, faces, ...), An incidence matrix k attaching k -cells to ( k - 1) -cells, with sparse entries ± 1 indicating orientation. A k -chain is a formal linear combination of k -cells. The incidence matrices map k -chains to ( k - 1) -chains, and satisfy k - 1 k = 0 . This defines a so-called chain complex . 0-form 1-form 2-form A discrete k -form assigns a degree of freedom to each oriented k -cell of a mesh. This defines a linear map on k -chains; the pairing between forms and chains is denoted by , ·i . The discrete exterior derivative is the coboundary operator d k := * k +1 . Therefore, h dα, σ i = h α, ∂σ i (Stokes’ theorem) holds automatically, as well as d k +1 d k = 0 . 1
Stern, Discrete Differential Forms for Numerical Relativity 24th PCGM, 3/22/08 Successful Application to Computational E&M Let the electromagnetic potential A be a discrete 1 -form. Then F = dA is stored on faces, while * F is stored on dual faces.

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