Numerical Solution of Partial Differential Equations, SS 2014 Exercise Sheet 10
Prof. Peter Bastian Deadline 02. Juli 2014
IWR, Universit¨at Heidelberg
EXERCISE1 CROUZEIX-RAVIARTA-POSTERIORI ERROR ESTIMATION
LetΩ R2 be a domain with Lipschitz boundary andtThuh be a family of conform and shape regular simplex triangulations with maximum edge sizeh. LetFh denote the set of all edges of the meshTh. For givenThwe define following spaces
Pc,01 tvh PC0pΩq, vh
BΩ0, @tPTh :vh
tPP1u and
Ppt,01 tvhPL1pΩq, @tPTh:vh
tPP1, @ePFh :
»
e
vvhwe0u, wherevfweis the jump of function f over edgee(ife BΩwe definef
e 0). Only space Pc,01 is a susbspace ofH01pΩq.
For spaceVh :H01pΩq Ppt,01 the bilinear formah:VhVhÑRis defined as
ahpuh, vhq ¸
tPTh
»
t
∇uh∇vhdx
together with the energy norm
}vh}Vh:a
ahpvh, vhq.
The functionuPH01pΩqanduh PVhare solution of
@vPH01pΩq:
»
Ω
∇u∇vdx
»
Ω
f vdx.
and
@vh PVh: ¸
tPTh
»
t
∇uh∇vhdx
»
Ω
f vhdx.
Show the a-posteriori error estimation:
}uuh}Vh¤cp¸
tPTh
etpuh, fqq inf
vhPPc,01 }uhvh}Vh,
where the constantcdepends only on the shape regularity of the mesh (independent ofh) and use etpuh, fq ht}f ∆uh}0,t
1 2
¸
ePBt
h
1
e2}vBnuhw}0,e,
withheandhtdenoting the length ofeand the longest edge inTh respectively. 6 points
EXERCISE2 CONVECTION-DIFFUSIONPROBLEM
Inuebungen/uebung10 of your dune-npdemodule you can find a program that solves a convection- diffusion problem
∇ pkpxq∇uq apxq ∇u0 xPΩ upxq gpxq, xP BΩD
papxq kpxq∇uq njpxq, xP BΩN
usingQ1andQ2finite elements on domainΩ r0,2s r0,2s R2.
1. First of all we will solve only a diffusion problem (apxq ~0) with boundary conditions u1forx1 0, u 1forx1 2, (Dirichlet on the left and on the right side)
∇un0otherwise.
The permeability field is heterogeneous. In the program we used permeability fieldKa. Your task is to implement permeability fieldKb, see picture.
k1 k2
k1 k2
p2,2q
p0,0q p2,0q
p0,2q
Ka
p2,2q
p0,0q p2,0q
p0,2q
Kb
2. Have a look at the functionflux(you can find it in the fileutilities.hh). What does the function compute?
3. Compare the results offluxfunction forKaandKbwith coefficientsk13.104, k2 101for different refinement level and polynomial degrees. Does it converge to some value?
4. Now we will add advection. Set a p1,0qT (parameter convection 1 in uebung10.ini and k1 k2 103. What do you observe in the solution for different grid refinements? Can you explain these phenomena?
6 points