IWR – Universit¨at Heidelberg Prof. Dr. Guido Kanschat Alec Gilbert
Due date:16.11.2018
Homework No. 5
Finite Elements, Winter 2018/19 Problem 5.1: Galerkin equations
Consider the one-dimensional problem
−u00+u=f inΩ = (0,1), for the spaceV =H01(Ω).
We define the equidistant mesh
xj=jh, j= 0, . . . , n, n+ 1 with h= 1 n+ 1
on the intervalΩwithn+ 1mesh cellsTj = (xj−1, xj). The finite-dimensional subspaceVn is now the span of piecewise linear functions
ϕj(x) =
x−xj−1
h , ifx∈(xj−1, xj],
xj+1−x
h , ifx∈(xj, xj+1),
0, otherwise,
forj= 1,2, . . . , n.
(a) Sketch the domainΩwith its subdivision and a reasonable number of functionsϕj.
(b) Argue that the spaceVncontains exactly all functions which are linear on each cellTj, continuous onΩand have zero boundary conditions.
(c) Set up the Galerkin equations.
(d) Compute the2×2cell matricesAkand cell vectorsbk.
(e) Forn= 4, assemble the cell matrices and cell vectors into the global matrixAand the global vectorb. What does the matrix look like for generaln?
(f) Calculate the solution of Galerkin equations for the right hand sidef(x) = 1and plot it forn= 4.
Problem 5.2: Integral node functionals
In two dimensions, a finite element on a triangle shall consist of the space of quadratic polynomialsP2, and shall utilise the node functionalsNidefined by
Ni(f) =f(Xi) i= 1,2,3, Ni(f) = 1
|Ei−3| Z
Ei−3
f(x) ds, i= 4,5,6.
Here,Eiis the edge of the triangle facing the vertexXi, and|Ei|is its measure.
(a) Show that this element is unisolvent.
(b) Derive a basis{ϕj}forP2such thatNi(ϕj) =δij.
