All right, welcome back.
We'll continue, where we left off with the previous segment,
was in identifying the strong form of the linear elliptic PDE, in one dimension.
We sort of sketched out,
an approach to generating analytic solutions to the strong form.
And then, I made the point that this is not a very general approach,
simply because it's not a very useful approach I should say.
Simply because as, as you know, the given data, and as boundary conditions get more
complex, in particular in multiple dimensions, as you can imagine.
Domains of interest in the context of many problems are not just a simple
regular domains, right?
And that introduces a further element of complexity,
to solving partial differential equations, in strong form.
Okay?
So, analytic solutions are limited,
and one tries to find approximate solutions.
Okay?
The first obvious way to approximate any differential equation is to
go to something like, a finite difference method, right?
Where a finite difference methods simply take any derivative, and
replace it with a difference.
Okay, and this has spawned a whole class a whole field, of finite difference methods.
What we try to do with finite element methods is, is a little different.
Okay, we take if a mathematical approach that is fundamentally, quite different.
And the basis of that difference in approaches,
is to go from the strong form to you guessed it, the weak form of the PDE.
Okay? So, that is the topic of this segment.
All right?
The weak form of a linear
Elliptic.
PDE in One dimension.
All right.
What I'm going to do is, first give you the weak form, and
then, develop more ideas about it.
Okay, so, the first part of this segment,
the next few minutes may seem a little formal, but don't worry,
by, by the end of before too long actually, you will be masters of it.
Okay, so, here is the weak form.
The weak form is the following.
It is to find U, okay?
U of x, belonging to S.
Now, S for us is a space of functions.
We talked about finding a function u.
And when we see that it belongs to S we are thinking of S as some sort of
collection of functions, right?
Some sort of space of functions,
from which we expect to draw our actual solution u.
Okay?
When I say space of functions, you may think of you know, any class of function
just to fix the ideas of polynomials, or, you know, specific types of polynomials.
Legendre polynomials, or Lagrange polynomials, or
you may not want polynomials.
You may want to have a harmonic functions, right?
Or, or exponential functions, or something.
Okay?
This is the sort of thing we, we have in mind when we say,
that u belongs to a space of functions S.
Okay? All right.
We want to see a little more about, what the space of function is?
Okay? And we say that the space of functions u,
[INAUDIBLE] S, okay?
Consists of all u, right?
Such that u at
0 equals u not.
Okay?
All right.
And.
So, what I'm doing here is sort of building in the Dirichlet boundary
condition, okay?
Into this sub-space of functions.
What we're saying here is that, we're only interested in solutions you would satisfy
the Dirichlet boundary condition.
Now of course, there is a possibility of having two Dirichlet boundary conditions.
When we saw the strong form, we observed that on the right-end of, of the domain,
at x equals L.
You could have either another Dirichlet boundary condition, or
a Neumann boundary condition.
It turns out to be a little cumbersome to develop the weak form, for both cases.
And so, I'm going to develop the weak form, for
a single Dirichlet boundary condition.
Okay? At x equals 0.
Later on, we will see what happens,
when we have Dirichlet boundary conditions at x equals 0 and x equals f, okay?
All right. If, if we did have that,
we would build that condition also into the space S, okay?
For now, we're just saying, that we have a single Dirichlet boundary condition, okay?
So what, so, let me write that.
We're assuming, we're, we're considering a case
where there Dirichlet boundary condition at holds, at x equals 0 only.
Okay?
Let me see actually, let's consider this case.
Okay?
Just, just, because otherwise, it just gets a little cumbersome to develop,
in most general form.
We'll come back to it, all right.
So, this is what [LAUGH] we want to do.
We want to find u, belonging to this particular space S.
Which for now, is completely general.
All we're saying is that, it needs to satisfy the Dirichlet boundary conditions.
Okay?
So, find u, belonging to space S given all the other data that we have, right?
Given u 0, t because we're developing it, with the single boundary condition, right?
So, we are not considering u g, now.
Okay.
So, we're given u 0 t.
We're given the function f, right?
Which is our forcing function,
our body force in the context of the elasticity problem.
And, the constitutive relation.
Sigma equals E U comma x, okay?
So, we're given all this data.
Which, which was the same as, as the case, for the strong form.
Okay?
However, there is more, okay?
Such that.
For all the w belonging to V.
Okay? Now, this is new.
All right?
This symbol is for all, okay?
Right? So now, we've done something new.
We've introduced a new function w, which was not in the mix at all.
We are saying that it, is a function belonging to some space V.
Okay, think of V as the same sort of concept our S is, right?
If you think of S being some plane kind of polynomials,
maybe V is the same sort of class of functions.
But we want to say a little more about V, okay?
So V, consists of all functions w such that, w
And 0 equals 0.
Okay?
So w, also satisfies Dirichlet boundary condition except, that it is homogenous.
Okay?
It is a homogeneous Dirichlet boundary condition.
Okay?
If we had a we, if we were considering a boundary value problem,
with two Dirichlet boundary conditions on u, at x equals 0 and x equals L.
We would likewise, have w at L also, equal to 0, okay?
Note that so far, w is something we've cooked up, we've just,
we've just conjured it up.
So, we are allowed to say, what we want about w.
Okay? All right.
So, let me read what we have, so far.
We're not yet done.
Find u belonging to S, where S is that,
given u not t f of x and, and the constitutive relation,
sigma equals E u comma x such that, for all w belonging to V,
for V is as specified the following holds now.
[INAUDIBLE] What holds?
Integral 0 to L.
W comma x, sigma d x
Equals integral
0 to L w f d x
plus W at L.
T, all right?
Essentially, what I've done is, go to an integral form.
This is our weak form, okay?
I want to do, just one more thing here.
Observe that I'm integrating over x, right?
And x really x I mean, integrating over x going from 0 to L.
Which is effectively our volume in one dimension, right?
Now, recognizing that though we're working in one d,
we're really thinking of problems that you know,
the canonical problem that we are considering here is that of elasticity.
Which actually has some cross sectional area, right?
A, right? Which could potentially be the function
of x, okay?
In order to make connection,
with what we're were going to do when we go to multiple dimensions, I want you to
take the step of simply multiplying everything here, through by A, okay?
Okay?
Because what that does is A d x then,
becomes A volume element, right?
Okay?
Likewise here.
All right.
And what we get here, is like a boundary force.
'Kay? This one made the,
the ultimate transition to three dimensions completely seamless.
Okay? All right.
And, and, A is a lot to be a function of x.
Okay, so, this really is our weak form of the equation.
Okay?
It's considerably different,
if you don't have experience with this type of equation,
it may look as though it bears no relation at all, to what we started with.
Right? To the strong form.
It does bear a relation, right?
In fact, we're going to show in a short time that it is
completely equivalent to the strong form.
This weak form by the way, is the basis for
approximations that leads to the finite element method.
It's a, it is, it is it is a fundamental aspect of a class of approximation,
or a, or a class of methods, which are called variational methods