Introduction to Numerical Methods for Variational Problems by Hans Petter Langtangen & Kent-Andre Mardal

Author:Hans Petter Langtangen & Kent-Andre Mardal
Language: eng
Format: epub
ISBN: 9783030237882
Publisher: Springer International Publishing

Starting with the Galerkin method,

integrating u″ψ i by parts results in

The first boundary term, u′(L)ψ i(L), vanishes because u(L) = D. The second boundary term, u′(0)ψ i(0), can be used to implement the condition u′(0) = C, provided ψ i(0) ≠ 0 for some i (but with finite elements we fortunately have ψ 0(0) = 1). The variational form of the differential equation then becomes

(5.20)

5.3.2 Boundary Term Vanishes Because of the Test Functions

At points where u is known we may require ψ i to vanish. Here, u(L) = D and then ψ i(L) = 0, . Obviously, the boundary term u′(L)ψ i(L) then vanishes.

The set of basis functions contains, in this case, all the finite element basis functions on the mesh, except the one that is 1 at x = L. The basis function that is left out is used in a boundary function B(x) instead. With a left-to-right numbering, ψ i = φ i, i = 0, …, N n − 2, and :

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