Math 272c: Numerical Differential Equations Instructor: Randolph E. Bank
Spring Quarter 2019
Homework Assignment #4 Due Friday, May 24, 2019
Exercise 4.1. Let τ be a triangle with vertices vi, 1 ≤ i ≤ 3 oriented in counterclockwise order. Let edge ei be the edge opposite vertex i. Let φi denote the barycentric coordinates (piecewise linear nodal basis functions). Let li denote the length of edge i and let ti denote the unit tangent vector. Let ni denote the unit outward normal vector and hi is height. This problem concerns some basic geometric relations among all of these quantities. For example, the area of the triangle τ can be computed from lihi = 2|τ|.
a. Prove the following simple results:
∇φi = −ni/hi
φ1 + φ2 + φ3 = 1 ∇φ1 +∇φ2 +∇φ3 =0 l1t1 +l2t2 +l3t3 =0
b. Let the reference element be the triangle τˆ with vertices be given by ν1 = (0,0)t, ν2 = (1,0)t, and ν3 = (0,1)t. Let the affine map from τˆ to τ be given by
v = Jν + v1 where J is a 2 × 2 matrix. Prove the following:
−l2t2 J−t = ∇φ2 ∇φ3
J = l3t3
c. Using the above identities prove:
l3tt3
a. Write down the standard nodal basis functions in terms of the barycentric coordinates φi.
l1tt1
l 2 t t2 ∇ φ 1
0 −1 1 ∇ φ 2 = 1 0 − 1
∇ φ 2
Exercise 4.2. Let Sh be a continuous, piecewise quadratic finite element space defined on
−1 1 0 a triangulation Th. Let τ ∈ Th denote an element in the triangulation.
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b.
Mathematics 272c
Let Vh be continuous piecewise linear finite elements and Wh the quadratic bump functions. Then we have the hierarchical decomposition Sh = Vh ⊕ Wh. Let u ∈ Sh. Then on element τ, u has the representation
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u = u(vi)φi(v) + αibi(v) i=1
where bi is the quadratic bump function for edge i. Prove α i = l 2i ∂ 2 u
2 ∂ t 2i
where ∂/∂ti denotes the directional derivative in the tangent direction ti. Hint: use
the identities from the first exercise.
Exercise 4.3. Let Sh be a continuous, piecewise cubic finite element space defined on a
triangulation Th. Let τ ∈ Th denote an element in the triangulation.
a. Write down the standard nodal basis functions in terms of the barycentric coordinates
φi.
b. Let Vh be continuous piecewise quadratic finite elements and Wh the hierarchical ex- tension, consisting of piecewise cubic polynomials that are zero at the Lagrange points for the piecewise quadratic elements. Then we have the hierarchical decomposition Sh = Vh ⊕Wh. Find local basis functions for Wh on element τ in terms of the barycen- tric coordinates φi. (There are several good choices).
c. Let u ∈ Sh. Then on element τ, u has the representation
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u = uiqi(v) + αibi(v)
i=1 i=1
where qi are the nodal quadratic basis functions and ui are the values at the quadratic Lagrange points (vertices and edge midpoints). The bi are the basis functions for Wh you found in part b above. For your choice of bi, determine the αi.