pro 2.4 in Numerical solution of partial differential equations FEM, Claes Johnson

truongpitt

I have though about it a lot, but did not get the solution. Can anyone help me with this. This is pro 2.4 page 63, in the book: Numerical solution of partial differential equations by the finite element method, written by Claes Johnson.

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chiro

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Re: pro 2.4 in Numerical solution of partial differential equations FEM, Claes Johnso

Hey truongpitt.

What have you tried? Can you show us any past working out?

truongpitt

Re: pro 2.4 in Numerical solution of partial differential equations FEM, Claes Johnso

Thank you, here is what I thought.

The left hand side is a line integral, the right hand side is something of double integral. Thus, the first thing I can think of is the relation ship between line integral and the double integral, I know the Green's formula or something similar to this. But I did not go any far.

chiro

MHF Helper
Re: pro 2.4 in Numerical solution of partial differential equations FEM, Claes Johnso

You will to first convert your line integral to a form of Adx + Bdy to use Greens theorem directly.

This will involve using a parameterization to express your v in terms of components with respect to infinitesimals dx and dy.

For a rectangle (axis aligned) you will get two terms that have Adx and two terms that Ady with different limits.

I have another suggestion though - Take a look at the banach norm and consider a new boundary that is circular that completely encloses the rectangle within it so that Gamma is a circle. Also remember that v is continuous and the boundary is finite as well because when this holds then the Banach Norm can be used since it is equivalent to things being continuous.

Now prove that there exists some constant where if you took the norm of the integral you could write it as C*||v|| for some C and use the fact that the original integral is going to be less than that.

truongpitt

Re: pro 2.4 in Numerical solution of partial differential equations FEM, Claes Johnso

But it is very vague to me. I do not have a good picture about what is supposed to do in your way. Can you write it clear, set up things for the problem, what are needed to do in your way?

chiro

MHF Helper
Re: pro 2.4 in Numerical solution of partial differential equations FEM, Claes Johnso

If a space is continuous then it is a Banach space. The Banach norm is ||AB|| <= ||A||*||B||.

Integrals have a norm provided you are in the space L^2 and the inner product within an integral is <a,b> = Integral ab wrt some single variable.

Basically the idea was to define a region that contained the square - but bigger (like a circle) and show that this is <= C*||v|| and use that to show the rectangle also has that property by subsets.

As an example if you have two functions L^2(R) then the inner product is <a(x),b(x)> = Integral [fixed boundary] a(x)b(x)dx.

expatmath

Re: pro 2.4 in Numerical solution of partial differential equations FEM, Claes Johnso

I have started to read Claes Johnson's book. Do you have solutions for 1.1 and 1.2?

Thank you.