Math 272b: Numerical Differential Equations Instructor: Randolph E. Bank
Winter Quarter 2019
Homework Assignment #1 Due Friday, January 18, 2019
Exercise 1.1. Consider the Stokes equations in two dimensions. The bilinear form is B(u, v) = a(u, v) − b(v, p) − b(u, q) = (f, v)
a(u,v)=
∇u·∇vdx= ∇u1∇v1 +∇u2∇v2dx ΩΩ
Ω
b(u, q) =
∇ · u q dx
foru,v,∈H01×H01 andp,q∈L02. Wediscretizeonashaperegular,quasiuniform,triangular mesh, using the mini-element: continuous piecewise linear finite elements with cubic bubble functions Pˆ1 × Pˆ1 ⊂ H01 × H01, and continuous piecewise linear finite elements with average value zero P ̃1 ⊂ L02.
1. On triangle t, let u be a linear polynomial and v be the cubic bubble function with support on t. Show:
∇u∇v dx = 0 t
is helpful. Here the ci are baricentric coordinates (linear basis functions) on t and v = 27c1c2c3. Remember c1 + c2 + c3 = 1 and ∇c1 + ∇c2 + ∇c3 = 0.
2. Show that the block 2 × 2 KKT system
A BtU F
B0P=0 can be written as a block 5 × 5 system
A1 0 0 0 B1TU1 F1
The identity
m!n!k!2|t|
c1 c2c3 = (m+n+k+2)!
m n k t
0 0 0
D 1 0 0
0 0 A 2 0
Bˆ 1T Uˆ 1 Fˆ 1
B 2T U 2 = F 2
D
Bˆ T Uˆ Fˆ 2222
0
B1 Bˆ1 B2 Bˆ2 0 P 0
where the Ai are associated with the piecewise linear parts of the velocity, and the Di are diagonal matrices associated with the cubic bubble functions.
2
3.
4.
Mathematics 272b
Use static condensation (i.e., block Gaussian elimination) to reduce this system to the block 3 × 3 system
A 1 0 B 1T U 1 F 1 0 A 2 B 2T U 2 = F 2 B1 B2 −C P G
where C is symmetric and positive semi-definite. This reduced Schur complement system is the one usually solved in practice. The bubble function part of the solution is not computed; the bubble functions were introduced mainly for stability and not for accuracy.
Show that the reduced system is actually a special case of the Petrov-Galerkin dis- cretization studied in class (i.e., the matrix C and the right hand side G are equal to the corresponding terms in the Petrov-Galerkin formulation for a particular choice of constants). Thus the mini-element and the Petrov-Galerkin formulation stabilize the Stokes equations in a similar fashion.