MATH 610
Homework #5
Finite Element Method for elliptic problems
(1) Consider the following problem: find u(x, y, z ) which satisfies
−∆u+u=−∂2u
∂x2−∂2u
∂y2−∂2u
∂z2+u= 1,in Ω,and u= 0 on Γ,
where Ω is a bounded domain in R3with a polygonal boundary Γ.
(a) (10 pts) Derive the variational formulation of this problem. For a partitioning
of Ω into tetrahedral finite elements define the finite dimensional space for the
Ritz method consisting of linear finite elements.
(b) (10 pts) For a finite element given by its vertices P1= (x1, y1, z1), P2=
(x2, y2, z2), P3= (x3, y3, z3), P4(x4, y4, z4), introduce the barycentric (volume,
homogeneous) coordinates (λ1, λ2, λ3, λ4).
(c) For a finite element given with its vertices P1,P2,P3,P4compute:
(a) (20 pts) the element ”stiffness” matrix;
(b) (20 pts) the element ”mass” matrix.
(2) (30 pts) Find in terms of the barycentric coordinates λjthe “nodal basis” for the FE
(τ, P,Σ) where:
(1) τa triangle, defined by its three vertices P1, P2, P3,
(2) Pis the set of polynomials of degree at most 2 on τ, and
(3) Σ is the set of linear functionals
Σ = nψ1(v) = v(P1), ψ2(v) = v(P2), ψ3(v) = v(P3),
ψ4(v) = 1
|e3|Ze3
v(s)ds, ψ5(v) = 1
|e1|Ze1
v(s)ds, ψ6(v) = 1
|e2|Ze2
v(s)dso.
Here we use the following notations for the sides of τ:e3=P1P2, e1=P2P3, e2=
P3P1.
(3) (30 pts) Find the “nodal basis” (in terms of the barycentric coordinates λj) for the
FE (τ, P,Σ1), where τand Pare as in the previous problem and Σ1is
Σ1=nψ1(v) = v(P1), ψ2(v) = v(P2), ψ3(v) = v(P3),
ψ4(v) = v(P1)−2v(P4) + v(P2),
ψ5(v) = v(P2)−2v(P5) + v(P3),
ψ6(v) = v(P3)−2v(P6) + v(P1)o.
(4) (30 pts) Let τbe a simplex in 3-D defined by its four vertices P1, P2, P3, P4. Let
λ1, λ2, λ3, λ4be the introduced above barycentric coordinates. Prove the following
formula (Bell-Holland):
Zτ
λm
1λn
2λp
3λq
4dx =m!n!p!q!
(m+n+p+q+ 3)! 6|τ|.
Hint. You can use induction in the dimension of the space.
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