Consider the finite element spaces Vh =nvh ∈ C0(Ω)2: vh ˜ T ∈ P12 for all ˜T ∈ Th 2 and vh=0 on ∂Ωo , Wh = ph ∈ C0(Ω): ph T ∈ P1 for all T ∈ Th and Z Ωphdx=0

Exercises to Wissenschaftliches Rechnen I/Scientific Computing I (V3E1/F4E1)

Winter 2016/17

Prof. Dr. Martin Rumpf

Alexander Effland — Stefanie Heyden — Stefan Simon — Sascha Tölkes

Problem sheet 12

Please hand in the solutions on Tuesday January31!

Exercise39 6Points

LetΩbe a polygonal domain with regular triangulationT_h. The triangulationTh 2 is obtained by adding new vertices at the midpoints of the edges of each element and joining the vertices correspondingly (see Figure1). Consider the finite element spaces

V_h =ⁿv_h ∈ C⁰(_Ω)²: v_{h ˜}

T ∈ P₁² for all ˜T ∈ Th

2 and v_h=0 on ∂Ωo

, W_h =

p_h ∈ C⁰(_Ω): p_h

_T ∈ P₁ for all T ∈ T_h and Z

Ωp_hdx=0

.

Figure1: TriangulationsT_h (thick) andTh 2.

(i.) Show that the discrete inf-sup conditon for Stokes problem is satisfied.

Hint: Follow the lines of Lemma4.14.

(ii.) Let (v,p) ∈ H^2,2(_Ω)×H^1,2(_Ω) be the solution to Stokes problem and (v_h,p_h) the associated discrete solution. Show the estimate

kv−v_hk_1,2,Ω+kp−p_hk_0,2,Ω ≤Ch(kvk_2,2,Ω+kpk_1,2,Ω).

Exercise40 4 Points Let ρ > 0, n ≥ 1, p,q ∈ [1,∞] and 0 ≤ m ≤ l be fixed. Consider a bounded domain Ω ⊂ _Rⁿ with polygonal boundary and T_h a triangulation of Ω such that ρh ≤diam(T) ≤hfor everyT ∈ T_h. Show that for anyvh ∈ Vh ⊂ H^l,p(_Ω)∩H^m,q(_Ω), whereV_h is a finite element space, and any T ∈ T_h the inequality

kvhk_l,p,T ≤Ch^{m−l+np−nq}kvhk_m,q,T

holds true, whereCdoes neither depend on v_hnor on h.

Exercise41 6Points

LetΩ = [0, 1]², M_h be a regular mesh onΩ composed of quadratic elements and V_h :=ⁿv_h ∈ L²(_Ω,R²) : v_h

K ∈ Q²₁ for all K∈ M_h, andv_h=0 on ∂Ωo , Wh :=

wh ∈ L²(_Ω,R) : wh

K ∈ Q₁for all K ∈ M_h, Z

Ωwh =0

. Recall thatQ₁was introduced in exercise 10. Consider the bilinear forms

a : V_h×V_h →_R, a(u_h,v_h) = Z

Ω

∑2 i=1

∇(u_h)_i· ∇(v_h)_idx, b: V_h×W_h →_R, b(v_h,q_h) = −

Z

Ω(divv_h)·q_hdx

associated with Stokes problem. Construct a counterexample showing that the inf-sup-condition is not satisfied, i.e.

q_hinf∈W_h sup

v_h∈V_h

|b(vh,qh)|

kv_hk_1,2,Ωkq_hk_0,2,Ω =0 .