TU München
Zentrum Mathematik
Lehrstuhl M16
E. Sonnendrücker
A. Ratnani
Wintersemester 2015/2016
Advanced Finite Element Methods
Exercise sheet 1
I. Lagrange Pk Finite Element for the 1D Laplace problem:
−u00 + µu = f in [0, 1]
µ≥0
(1)
A. Variational formulations:
1) Write the variational formulation for homogeneous Dirichlet boundary conditions u(0) =
u(1) = 0, and prove that this variational formulation admits a unique solution.
2) Consider the Dirichlet boundary conditions u(0) = 0, u(1) = 1. Find a smooth function satisfying these boundary conditions. Write the variational formulation for these
boundary conditions, and prove that this variational formulation admits a unique solution.
3) Consider the Neumann boundary conditions u0 (0) = α, u0 (1) = β (α, β ∈ R). Write
the corresponding variational formulation. Show that it admits a unique solution for
µ > 0. What happens in the case µ = 0.
B. Consider the Lagrange Finite Element ([a, b], Pk ([a, b]), Σ = {p 7→ p(xj ), 0 ≤ j ≤ k}), with
a = x0 < x1 < · · · < xk = b.
1) Draw the degrees of freedom and show that the Finite Element is unisolvant.
2) Define the approximation space Vh based on this Finite Element.
3) Express the different matrices involved in the Finite Element approximation with respect to integrals over the basis functions and the data.
4) Compute the element matrix on the reference element [−1, 1] assuming that the degrees
of freedom are at the Gauss-Lobatto points. Gauss-Lobatto quadrature can be used
for the integrals. Comment on the quadrature error.
5) Express the coefficients of the global matrices with respect to the reference element
matrices.
II. Lagrange Qk Finite Element for the 2D Laplace problem
−∆u + µu = f in [0, 1] × [0, 1]
µ > 0,
u = 0 for x = 0 and x = 1,
∂u
= g for y = 0 and y = 1.
∂n
(2)
(3)
(4)
A. Write the variational formulation and prove that this variational formulation admits a
unique solution.
B. Let K = [−1, 1] × [−1, 1]. Consider the Lagrange Finite Element (K, Qk (K), Σ = {p 7→
p(xi , xj ), 0 ≤ i, j ≤ k}), with −1 = x0 < x1 < · · · < xk = 1.
1) Show that the Finite Element is unisolvant, and that a basis is given by ϕi,j (x, y) =
lk,i (x)lk,j (y), 0 ≤ i, j ≤ k, where lk,i (x) denotes the ith Lagrange interpolation polynomial of degree k at the points x0 , x1 , . . . , xk .
2) Define the approximation space Vh based on this Finite Element.
3) Express the different matrices involved in the Finite Element approximation with respect to integrals over the basis functions and the data.
4) Compute the element matrix on the reference element [−1, 1]×[−1, 1] assuming that the
degrees of freedom are at the Gauss-Lobatto points. Verify that the element matrices
can be expressed using the 1D element matrices of the previous exercise.
5) Express the coefficients of the global matrices with respect to the reference element
matrices.
III. Finite element approximation of the biharmonic problem
∆2 u + u = f in Ω, u = 0 and
∂u
= 0 on ∂Ω.
∂n
(5)
A. Variational formulation
1) Using the Green formula for the Laplace operator and the Green formula for the divergence operator, derive the Green formula for the biharmonic operator ∆2 .
Hint: notice that ∆2 u = ∇ · ∇∆u.
2) Derive the variational formulation.
B. We consider the finite element (K, P5 (K), Σ), where K is a triangle of vertices (a1 , a2 , a3 ),
P5 (K) is the space of polynomials of degree at most five and Σ contains the following linear
forms
p 7→ p(ai ), i = 1, 2, 3,
p 7→
p 7→
∂2p
(ai ),
∂x2
∂p
(ai ),
∂x
p 7→
p 7→
∂p
(ai ),
∂y
∂2p
(ai ),
∂x∂y
p 7→
i = 1, 2, 3,
∂2p
(ai ),
∂y 2
i = 1, 2, 3,
∂p
(mi ), i = 1, 2, 3,
∂n
where mi is the midpoint of the edge opposite to ai . This element is called Argyris triangle.
p 7→
1) What is the dimension of P5 (K) ?
2) Show that if p ∈ P5 (K) is such that σ(p) = 0 ∀σ ∈ Σ, then λ21 λ22 λ23 divides p where λi
is the barycentric coordinate associated to the vertex ai .
3) Show that (K, P5 (K), Σ) is a finite element.
4) Define the discretisation space Vh associated to this element.
5) Show that Vh ⊂ C 1 (Ω̄).