The vectorsuPV anduhPVhfulfill apu, vq lpvq, @v PV and apuh, vhq lpvhq, @vhPVh

Numerical Solution of Partial Differential Equations, SS 2014 Exercise Sheet 7

Prof. Peter Bastian Deadline 11. Juni 2014

IWR, Universit¨at Heidelberg

EXERCISE1 HOMOGENEOUSDIRICHLET PROBLEM WITHP^{1 ELEMENTS}

Let Ω ra, bs € Rbe a real 1D domain and T_N be a equidistant grid on Ωwith grid sizeh pbaq{N forN PN. Let

V tv PH¹pΩq |vpaq vpbq 0u be a vector space and

Vh tvhPC⁰pΩq | @sPT :vh

s PP¹psq ^ vhpaq vhpbq 0u

be a finite-dimensional subspace. In addition letlbe a continuous linear forml :V Ñ Rand define a bilinear form

apu, vq

»

Ω

∇u∇vdx.

The vectorsuPV andu_hPV_hfulfill

apu, vq lpvq, @v PV and

apu_h, v_hq lpv_hq, @v_hPV_h. 1. Show, thatp,qV ap,qinduces a scalar product onV. 2. Show, thatupa ihq u_hpa ihqforiP0, . . . , N.

7 points

EXERCISE2 LOCALP¹BASIS ON REFERENCE ELEMENT

Letd 2. We consider a unit triangleT0 with nodes n0 p0,0q,n1 p1,0q,n2 p0,1qand an arbitrary triangleT €R² with nodesa₀ px₀, y₀q,a₁ px₁, y₁q,a₂ px₂, y₂q, see picture.

1.

The linear functionpPP1pT0qonT0 can be defined using values in pointsn_iand

the definition is unique. Find a node-basisϕ˜i, i1,2,3ofP1pT0q

fulfillingϕ˜_ipn_jq δ_ij.

px1, y1q

px2, y2q px3, y3q

T

x y

µT

p0,0q p1,0q p0,1q

T0

ξ η

2. Find a reference affine mappingµT :T0ÑT. Is this mapping unique and invertible?

3. The functionsϕ_i, i1,2,3are given by

ϕipξ, ηq:ϕ˜ipµ_T¹pξ, ηqq. Prove, thatϕPP1pT0qandϕipajq δi,j.

4. If you want to integrate a functionvd P P1pTq on theT, you can first integrate it on the refe- rence elementT₀(no change in quadrature points) and the result should be modified (regarding original element). Which factor should stay in front of the second integral?

»

T

vpx, yqdxdy . . .

»

T0

vpµ_Tpξ, ηqqdξdη.

5 points

EXERCISE3 ELLIPTIC OPERATOR INPDELAB

In this exercise you will solve a PDE for the first time. The programm in the directoryuebungen/ue- bung08of the actualdune-npdemodul solves a Laplace problem

∆upxq 0 xPΩ upxq gpxq xP BΩ

with Dirichlet boundary conditions usingP^kfinite elements and conforming triangulation mesh on domainΩ r0,2s r0,2s €R².

Your task is to modify thelocal operatorto solve a problem ∇pkpxq∇upxqq fpxq, xPΩ,

upxq gpxq, xP BΩ_D, kpxq∇upxq npxq jpxq, xP BΩ_N for scalar functionskpxq, fpxq, jpxq.

Create a VTK-file output and computeL₂-norm of the solution for the situation described below.

1. Choose Dirichlet and Neumann boundary parts of domainBΩ_D tpx, yq|x 0_x 2uand BΩN BΩzBΩD. Letgpx, yq xbe a function on Dirichlet boundary andjpx, yq 0be a zero Neumann flux function. The functionkis not continuous and is defined by

kpx, yq

$' '' '&

'' ''

%

1 x¤1^y¡1 10⁵ x¡1^y¡1 1 x¡1^y¤1 10⁵ x¤1^y¤1

and we have no source term (f 0).

2. Describe (qualitatively) properties of the solution and compare it to the original solution. How does it change if the functionjdoes not disappear, e.g.jpx, yq 1.

3. Describe (qualitatively) properties of the solution if we consider source-term fpx, yq exp^{4ppx1q2 py1q2q}

andjpx, yq 0.

10 points