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Additional info for Advances in Computers, Vol. 24
In fact, in this case, the action of form L on any function v belonging to HΓ11 (Ω) is expressed as: L(v) ≡ Ω f · v dΩ + Γ2 g · v dΓ , ∀v ∈ HΓ11 (Ω) . 96) v L2 (Γ2 ) L2 (Γ2 ) . 97) Thus, a new difficulty results from the application of the g Neumann condition defined on the Γ2 boundary. Since, control of L(v) should be performed only in relation to norm H 1 (Ω) of function v. This is why the term resulting from the Neumann condition and providing a measure of v for norm L2 (Γ2 ) should consequently be modified.
2) The three linear forms defined below are considered: σ1 : p → p(M1 ), σ2 : p → p(M2 ), σ3 : p → p(M3 ) . 42) 3) The identification of the three functions of canonical basis (p1 , p2 , p3 ) corresponds to the three barycentric functions (λ1 , λ2 , λ3 ) whose existence is established in the work of Daniel Euvrard . Though, it is pointed out that, by definition, the polynomial functions of degree less than or equal to one for the pair (x, y) prove the canonical property: σ j (λi ) ≡ λi (M j ) = δi j .
47) 3) The construction of the functions of canonical basis (p1 , p2 , p3 , p4 , p5 , p6 ) is worked out in the following way: Function p1 may be taken as an example. This second degree polynomial in relation to the pair (x, y) must be zero at the points below: M2 , M3 , M12 , M13 and M23 . Therefore, polynomial p1 , whose trace is a second degree trinomial of the oblique variable defining the parameter of segment M2 M3 , is identically zero over segment M2 M3 , being zero at the three points M2 , M3 and M23 .