In a Finite Element context, say I have a 1D finite element with 2 nodes, $x=0$ and $x=1$. The typically used shape funcions are
$N_1(x) = 1 - x$
$N_2(x) = x$
Would it make sense to use second order polynomials on that same element (without introducing an extra node at $x=0.5$) as described below?
$N_1(x) = 1 - 0.5x - 0.5x^2$
$N_2(x) = 0.5x + 0.5x^2$
Should I expect better accuracy? Is there something wrong in using those shape functions?
If $A_1(x)=1−0.5x−0.5x^2$ and $A_2(x)=0.5x+0.5x^2$. The interpolation fails with $u(x)=\sin(x)$ since the error estimation $$\|u(x)−u(0)A_1(x)−u(1)A_2(x)\|_{\infty}≈0.1644.$$ The linear interpolation $L_1(x)=1-x$ and $L_2(x)=x$ yields $$\|u(x)−u(0)L_1(x)−u(1)L_2(x)\|_{\infty} \approx 0.06$$
The usual quadratic interpolation $N_1(x)=2(x-1)(x-1/2)$, $N_2(x)=-4(x-1)x$, $N_3(x)=2x(x-1/2)$ gives, $$\|u(x)−u(0)N1(x)−u(1/2)N2−u(1)N3\|_{\infty}≈0.0072$$ Even doesn't interpolate well $u(x)=x^2$ where the usual quadratic interpolation is exact.