Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 63

Fig. 14. Nonuniform triangulations with a reentrant vertex

can be applied in every element T j to give

φw i dxdy = φ∂ i vdxdy

T j

j

φv ν i ds .

= −

∂ i φv dxdy +

( 5 . 2 )

T j ∂T j

j

Since v was assumed to be continuous, the integrals over the interior edges cancel.
Moreover, φ vanishes on ∂ , and we are left with the integral over the domain

−

∂ i φv dxdy.

By Definition 1.1, w i is the weak derivative of v .
(2) Let v ∈ H 1 () . We do not establish the continuity of v by working
backwards through the formulas (although this would be possible), but instead
employ an approximation-theoretical argument. Consider v in the neighborhood
of an edge, and rotate the edge so that it lies on the y -axis. Suppose the edge
becomes the interval [ y 1 − δ, y 2 + δ ]onthe y -axis with y 1 <y 2 and δ> 0. We
now investigate the auxiliary function

y 2

ψ(x) : = v(x,y)dy.

y 1

First, suppose v ∈ C ∞ () . It now follows from the Cauchy-Schwarz inequality
that

∂ 1 vdxdy

x 2 y 2

2

2

| ψ(x 2 ) − ψ(x 1 ) | =

x 1 y 1

1 dxdy ·| v |

x 2 y 2

2
1 ,

≤

x 1 y 1

2

≤| x 2 − x 1 |·| y 2 − y 1 |·| v |

1 , .