Problem 5.2: Trace inequality for polynomials (4 points)

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

Due date:29.11.2013

Homework No. 5

Numerical Methods for PDE, Winter 2013/14 Problem 5.1: Transformation of Quadrilaterals (6 points)

Transformation from the reference squareQb= [0,1]²to a general quadrilateral given by verticespⁱ= (xi, yi)^Tfori= 1, . . . ,4 can be obtained by the mappingF with

F(ξξξ) =p¹(1−ξ)(1−η) +p²(1−η)ξ+p³η(1−ξ) +p⁴ξη.

Here,ξξξ= (ξ, η)^T. The order of vertices follows the scheme

p¹ p²

p⁴ p³

Q

(a) Show that indeedQ=F(Q).b (b) Compute∇F(ξξξ).

(c) Compute the Jacobi determinantJ(ξ)and show thatJ(ξ)≥0, if and only if the quadrilateral is convex.

(d) Compute the eigenvalues of(∇F(ξξξ))^T∇F(ξξξ)and relate them tok∇F(ξξξ)kandk∇(F(ξξξ))⁻¹k.

(e) What happens toJ(ξ),k∇F(ξξξ)kandk(∇F(ξξξ))⁻¹katp¹ifp²gets close top¹?

Problem 5.2: Trace inequality for polynomials (4 points)

LetKbe a shape regular triangle of diameterh. Show that for any polynomialp∈Pkholds

kpk∂K ≤Ch⁻¹²kpkK

kpk∂K ≤Ch¹²k∇pkK

with constantsCdepending on the degreekand the constant of shape regularity.

Problem 5.3: Connection between the Shape of Triangles and the Stiffness Matrix (4 points)

We discretize the Poisson problem by piecewise linear finite elements on a triangulation of the unite square. Consider an arbitrary triangleTin this triangulation.

p¹ p²

p³

T

The nodal basis functionsϕihave the propertiesϕi(pj) =δij,i, j= 1,2,3.

(a) Show that

(∇ϕi,∇ϕi)_T >0 and (∇ϕi,∇ϕj)_T ≤0, i6=j as long as all interior angles are equal or smaller than ^π₂.

(b) The entries of the stiffness matrix are given by

aij = X

T∈Th

(∇ϕi,∇ϕj)_T.

Conclude from the first part of the exercise that the diagonal entries of the matrix are always positive and the off-diagonal entries are smaller or equal to zero.

(c) Bonus (3 points):Show that furthermore the conditions X

j6=i

|aij| ≤a_ii, X

j6=i^?

|ai^?j|< a_i?i^? for a fixedi^?

are fulfilled. Discuss the properties of the matrix.