APM4810
ASSIGNMENT 3 2024
UNIQUE NO.
DUE DATE: 18 OCTOBER 2024
, APM 4810
Assignment 3 2024
Unique Number:
Due Date: 14 October 2024
An Introduction to the Finite Element Method
Problem 1.20 Marks
Problem Statement: Assume that we have a domain Omega Ω in the form of a square
Ω=(0,2π) ^2, and suppose g∈H0^1(Ω) is a weak solution of −Δg=f with f∈L2(Ω). Using
Problem 1.16 (in the prescribed book), show that Δg∈L^2(Ω), and then use the Cauchy-
Schwarz inequality to show that all second derivatives lie in L^2, and thus ggg is an H^2
function.
Solution Outline:
1. Start with the weak solution property: Given that ggg is a weak solution of
−Δg=f, you have:
∫Ω∇g⋅∇ϕ dx=∫Ωfϕ dx,∀ϕ∈H01(Ω).
Here, ϕ\phiϕ is a test function from the Sobolev space H01(Ω).
2. Use Problem 1.16 (from the prescribed book): In Problem 1.16, it is likely
shown that if f∈L^2(Ω) and g∈H01(Ω), then Δg∈L^2(Ω).
By definition, Δg∈L2(Ω) means:
∫Ω∣Δg∣^2 dx<∞.
Use the Cauchy-Schwarz inequality: To show that all second derivatives of
ggg lie in L^2(Ω), start by considering the Cauchy-Schwarz inequality: