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Unformatted text preview: Discrete Differential Forms for Numerical Relativity Ari Stern Applied & Computational Mathematics California Institute of Technology astern@acm.caltech.edu 24th PCGM March 22, 2008 Stern, Discrete Differential Forms for Numerical Relativity 24th PCGM, 3/22/08 Discrete Differential Forms To specify an ndimensional mesh, we are given the following topological data for all k = 0 , 1 , . . . , n : • A list of oriented kcells (vertices, edges, faces, ...), • An incidence matrix ∂ k attaching kcells to ( k 1)cells, with sparse entries ± 1 indicating orientation. A kchain is a formal linear combination of kcells. The incidence matrices map kchains to ( k 1)chains, and satisfy ∂ k 1 ∂ k = 0 . This defines a socalled chain complex . 0form 1form 2form A discrete kform assigns a degree of freedom to each oriented kcell of a mesh. This defines a linear map on kchains; the pairing between forms and chains is denoted by h· , ·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 1form. Then F = dA is stored on faces, while * F is stored on dual faces.faces....
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This note was uploaded on 02/07/2011 for the course PHYS 101 taught by Professor Aster during the Spring '11 term at East Tennessee State University.
 Spring '11
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