Let xk be the vertices of a triangulation ofΩwithxk a °k i1hi,k0

Numerical Solution of Partial Differential Equations, SS 2014 Exercise Sheet 8

Prof. Peter Bastian Deadline 18. Juni 2014

IWR, Universit¨at Heidelberg

EXERCISE1 STIFFNESSMATRIX

We want to solve homogenuous Laplace equation ∆uf inΩ

u0 onBΩ withP1elements on the following grid:

NW

W S

Z N

SO 1 O

2

2 1 2

1

h

h

Basis functions of all inner nodes look the same and therefore all rows of the stiffness matrix are identical (except for boundary nodes). As a consequence it is sufficient to look at only one nodeZ.

LetN, O, SO, S, W, N W denote the neighbours ofZ.

Determine the matrix values of one row of the stiffness matrix corresponding to a inner node. In oder to do that you have to choose a numeration of basis functions. Express your solution asfinite difference stencilanalogue to finite difference methods.

5 points

EXERCISE2 BRAMBLE-HILBERT IN1D

Let Ω ra, bs € R, w : Ω Ñ R be a function with w P H²pΩq. Let xk be the vertices of a triangulation ofΩwithx_k a °_k

i1hi,k0. . . N andh_k ¡0such thatx0 aandx_N bholds.

Letvbe a piecewise linear interpolation ofwfullfillingvpxiq wpxiqforpi0. . . Nq. LetΩˆ r0,1s be the reference element andµ_k : ˆΩ Ñ rx_k1, x_ksbe the corresponding transformation to grid cell rx_k1, x_ks.

Show that forepxq:wvandˆe_kpxˆq:epµ_kpxˆqqit holds

|ˆek|_1,Ωˆ ¤ }Bxˆˆxeˆk}_0,Ωˆ

and withh max

1¤k¤Nth_kuit holds

}e}²1,Ω¤h²ph 1q}Bxxw}²0,Ω.

5 points

EXERCISE3 CONVERGENCERATES FORPOISSONEQUATION

LetΩ r0, as r0, bs €R², 0 a, bPR. The Poisson equation ∆upx, yq

3b

2y²b² 2yy³

p6x3aq 3a

2 x²a² 2 xx³

p6y3bq, px, yq PΩ (1) with homogenuous Dirichlet boundary condition has the analytical solution

upx, yq xypaxqpbyqa 2 x

b 2y

.

Inuebungen/uebung09of yourdune-npdemodule you can find a program that solves Poisson equa- tion (1) withP^kfinite element on a conform trianglular grid (UGGrid) and withQ^kfinite element on a conform quadrilateral gride (YaspGrid). As domain we choseΩ r0,2s r0,2s €R².

1. Implement a methodevaluatein the classExactGradient. This function should evaluate the gradi- ent ofu. Your program can than determine the norms}uu_h}0,Ωand}∇puu_hq}0,Ω. Extend the program such that it calculates and prints}uu_h}1,Ωand}uu_h}L₈pΩqand the corresponding convergence rates.

2. Plot}uu_h}0,Ω,}uu_h}1,Ωand}uu_h}L₈pΩqagainst the number of degrees of freedom forP^k andQ^kelements,k1,2. Use logarithmic scale on both axes.

3. Implement a function that calculates fpu_h,Ωq max

i |upa_iq u_hpa_iq|, wherea₀, . . . , a_N1 PΩare the vertices of our grid.

What does this function return forP¹ elements? Can you explain your observations?

Hint: Because this problem gets quite large you can compile your code with

make CXXFLAGS=’-O3 -march=native -g0 -funroll-loops -ftree-vectorize’

for faster execution.

10 points