IWR – Universit¨at Heidelberg Prof. Dr. Guido Kanschat Alec Gilbert
Due date:30.11.2018
Homework No. 7
Finite Elements, Winter 2018/19 Problem 7.1: H
1- and L
2-Error Estimates
Consider the following problem
−∆u+u=f, inΩ, u= 0, on∂Ω, whereΩis a convex polygonal domain inR2.
We will use a conforming FE method with piecewise linear finite elements (Vh⊂V) to approximate the above problem.
(a) Formulate the variational equation and its approximation, and name the solution spaceV as well as the ansatz spacesVh. (b) Discuss unique existence of the exact solutionu∈V as well as of the discrete solutionsuh∈Vh.
(c) Derive the energy error estimate
k∇(u−uh)kL2≤chkfkL2,
by assuminguhas elliptic regularity.
(d) Derive theL2-error estimate
ku−uhkL2 ≤ch2kfkL2
by Aubin’s and Nitsche’s duality argument.
Hint: For a given pde in weak form ”Findu∈ V, s.t. a(u, ϕ) = (f, ϕ)∀ϕ∈ V” the dual problem is given by ”Find z∈V, s.t.a(v, z) =J(v)∀v∈V”, where the error functionalJ(·)can be defined appropriately to the specific target.
