-A Posteriori Error Estimate (8 points)

IWR – Universit¨at Heidelberg Prof. Dr. Guido Kanschat

Due date:20.12.2013

Homework No. 8

Numerical Methods for PDE, Winter 2013/14 Problem 8.1: Dual Problem and Error Functionals (8 points)

Consider the following convection-diffusion equation

−ε∆u+β· ∇u=f inΩ, u= 0 on∂Ω,

with a sufficiently smoothβandε >0. The domainΩis a convex polygon in two space dimensions.

(a) Formulate the dual problem weakly as well as classically for a general linear error functionalJ(·).

(b) Draw the transport fields of the dual solution for the specific transport fields (i) β1= (1,1)^T,

(ii) β2= (y−0.5,0.5−x)^T.

(c) Formulate appropriate error functionalsJ(·)for the computation of the (i) energy errork∇(u−u_h)kL²(Ω),

(ii) L²-errorku−u_hkL²(Ω), (iii) the mean value of the solutionu.

Problem 8.2: “Sharp” L

-A Posteriori Error Estimate (8 points)

Consider the standard Poisson problem

−∆u(x) =f inΩ, u(x) = 0 on∂Ω, whereΩis a convex polygonal domain inR².

The a posteriori error estimate inL²is given by

kehk_L2(Ω)≤η_L2(uh) :=c X

T∈Ωh

h⁴_T ρT(uh)²+h⁻¹_T ρ∂T(uh)²

!¹₂ ,

withe_h=u−u_h, and

ρT(uh) :=kf+ ∆uhk_L2(T), ρ∂T(uh) =1

2k[∂nuh]k_L2(∂T), where[·]denotes the jump of the normal derivative on the edge of neighboring cells.

(a) Prove the estimate

k∂nuk²_L2(∂T)≤c

hTk∆uk²_L2(T)+h⁻³_T kuk²_L2(T)

,

by proving a similar estimate on the reference element.

Hint:Use the regularity theory for elliptic PDE’s and the trace equation.

(b) Use the estimate to prove thatη_L2(uh)is reliable and efficient in the following sense ηL²(uh)≤c1kehk_L2(Ω)+c2h²kfk_L2(Ω)

withh:= maxT∈Ω_hhT.

Each problem 5 points