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Mathematics
Differential Equations
Finding the weak form of an eq. with DG elements (finite elements)
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[QUOTE="Paul Colby, post: 6846589, member: 584221"] My understanding the weak form of a differential equation is obtained by multiplying the equation through by a set of testing functions then integrating this product over each element. The Galerkin method is to choose the testing functions and expansion functions as the same set of functions. The functional I gave will yield the Galerkin system matrix less the boundary conditions (AKA the right hand side vector). In this functional, ##\sigma##, the conductivity of the materials, is specified. No derivatives of ##\sigma## appear in the functional so there are no restrictions on continuity. Also required is that the potential, ##V##, be continuous and differentiable throughout the region. For any real materials, this is exactly what Maxwell's equations dictate. Failing to do this would spawn infinite ##E##-fields and we wouldn't want that to happen. So, I don't follow the physics of what you intend interior to the region. Now, I didn't give the part of the functional that yields the boundary conditions. I believe these are what account for the extra terms seen in the python code. They appear to be sums over the boundary and likely yield a properly tested boundary value equations. For specifying the value of ##V## on the boundary things are conceptually simpler. How this is done in FEniCSx? [/QUOTE]
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Mathematics
Differential Equations
Finding the weak form of an eq. with DG elements (finite elements)
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