Finite element method: Difference between revisions

Revision as of 00:41, 1 December 2003

Revision as of 00:43, 1 December 2003

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for all ''v'' in V. From now on, we will use ∫_T for the double integral ∫₀¹∫₀¹. One can see, via [[integration by parts]] that the last equality is equivalent to:

:ψ(''u'',''v''):=-∫_T(''u_xv_x''+''u_yv_y'') = -∫_T''u'g·v'' = φ_''v''(''g'')

where ''u' '' is the gradient of u, ''v' '' is the gradient of v and · is the Euclidian scalar product. The function ψ of ''u'' and ''v'' is in fact [[bilinear]], and it is the bilinear form associated with L. The functions ''v'' are called ''test functions''.

We note here that ψ only uses first derivatives, and it would therefore be possible to discuss solutions to the original problem while only assuming first derivatives. Also note that in fact in most cases, the solution ''u'' will be infinitely differentiable.