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2.4 Weak formulation

We restrict the support of our basis functions to the reference triangle and continue by formulating the continuity equation (1.5) in the weak form [Bra07]. Using the basis functions as a particular set of test functions, this yields

∫ ∫ ∫ ˆUt(ξ,η,t)φj(ξ,η)+ ∇ ⋅F(ˆU (ξ,η,t))φj(ξ,η) = S (Uˆ(ξ,η,t))φj(ξ,η). T T T
(2.7)

Applying the Gauss divergence theorem yields

∫ ∫T ˆUt(ξ,η, t)φj(ξ,η) Time derivative - ∮T F(ˆU (ξ,η,t))⋅∇ φj(ξ,η) Spatial derivative + dT F (ˆU(ξ,η,t))φj(ξ,η)⋅⃗n(ξ,η) Flux = ∫ S(ˆU (ξ,η,t))φ (ξ,η) Source T j
(2.8)

with ⃗n(ξ,η) the normal on each edge. Expanding Û, this yields the Ritz-Galerkin formulation

∫ ∑ Ui(t) ∫T i∑ dt φi(ξ,η)φj(ξ,η) Time derivative - ∮T F (∑ iUi(t)φi(ξ,η))⋅∇ φj(ξ,η) Spatial derivative + ∫dT F (∑ iUi(t)φi(ξ,η))⋅⃗n(ξ,η)φj (ξ,η) Flux = T S( iUi(t)φi(ξ,η ))φj(ξ,η) Source.
(2.9)

The following sections continue with successive discretization of those terms, aiming at a matrix and vector representation of all terms, see [GW08].

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