The rest of the numerical experiments shown in chapters 6 and 7 are performed on trial spaces LhL defined as in (2.5) and generated by Hermite elements of order r = 3. The associated basis functions over two contiguous segments in the mesh are explicitly given by
pj(x) =
−(x−xj−1)2(xj−1−3xj+2x)
(xj−xj−1)3 for xj−1 6x 6 xj (x−xj+1)2(xj+1−3xj+2x)
(xj+1−xj)3 for xj 6x 6 xj+1
0 otherwise
and
qj(x) =
(x−xj)(x−xj−1)2
(xj−xj−1)2 for xj−1 6x 6 xj (x−xj)(x−xj+1)2
(xj+1−xj)2 for xj 6x 6 xj+1
0 otherwise
Appendix A: Finite element spaces
Figure A.2: Cubic Hermite basis functions.
where h = 2Ln, −L = x0 and xn = L. The mass A0, stiffness A1 and bending A2
matrices are obtained as follows. Set
Bℓjk =
aℓ(pj, pk) aℓ(pj, qk) aℓ(qj, pk) aℓ(qj, qk)
and Bℓ = [Bℓjk]njk=1.
Then
Aℓ =
aℓ(q0, q0) · · · aℓ(q0, qn+1)
...
B ℓ
...aℓ(qn+1, q0) · · · aℓ(qn+1, qn+1)
.
The entries can be found explicitly from the tables below.
Appendix A: Finite element spaces
Appendix A: Finite element spaces
Appendix A: Finite element spaces
Appendix A: Finite element spaces
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