M1 Mathematics, 2011-2012
MM26E Numerical Approximation of PDEs Hervé Le Dret
$Q_3$-Hermite approximation
Below are the graphs of various basis polynomials of $Q_3$-Hermite interpolation on the unit square $[0,1]^2$.
$(x_1,x_2)\mapsto (1+2x_1)(1-x_1)^2(1+2x_2)(1-x_2)^2$
$(x_1,x_2)\mapsto (1+2x_1)(1-x_1)^2x_2(1-x_2)^2$
$(x_1,x_2)\mapsto x_1(x_2-1)(1-x_1)^2x_2^2$
And the corresponding $C^1$ hat functions (compare with the $Q_2$ only $C^0$ hat functions), constructed by patching together four copies of each of the shape functions above on four contiguous rectangles:
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