Weak formulation

Weak formulation

Weak formulations are an important tool for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, an equation is no longer required to hold absolutely (and this is not even well defined) and is instead has weak solutions only with respect to certain "test vectors" or "test functions".

We introduce weak formulations by a few examples and present the main theorem for the solution, the Lax–Milgram theorem.

General concept

Let V be a Banach space. We want to find the solution u in V of the equation

:Au = f,

where A:V o V' and fin V'.

Calculus of variations tells us that this is equivalent to finding uin V such thatfor all vin V holds:

: [Au] (v) = f(v).

Here, we call v a test vector or test function.

We bring this into the generic form of a weak formulation, namely, find uin V such that

: a(u,v) = f(v) quad forall vin V,

by defining the bilinear form

:a(u,v) := [Au] (v).

Since this is very abstract, let us follow this by some examples.

Example 1: linear system of equations

Now, let V = mathbb R^n and A:V o V a linear mapping. Then, the weak formulation of the equation

:Au = f

is: find uin V such that for all vin V the following equation holds:

: (Au,v) = (f,v).

Since it is sufficient in mathbb R^n to test with basis vectors, we get

: (Au,e_i) = (f,e_i) quad i=1,ldots,n .

Actually, expanding u=sum_{j=1}^n u_je_j, we obtain the matrix form of the equation

:mathbf A mathbf u = mathbf f,

where a_{ij} = (Ae_j, e_i) and f_i = (f,e_i).

The bilinear form associated to this weak formulation is

: a(u,v) = mathbf v^Tmathbf A mathbf u.

Example 2: Poisson's equation

Our aim is to solve Poisson's equation

:-Delta u = f,

on a domain Omegasubset mathbb R^d with u=0 on its boundary,and we want to specify the solution space V later. We will use the L^2-scalar product

:(u,v) = int_Omega uv,dx

to derive our weak formulation. Then, testing with differentiable functions v, we get

: - int_Omega Delta u v ,dx = int_Omega fv ,dx .

We can make the left side of this equation more symmetric by integration by parts using Green's identity:

: int_Omega abla u cdot abla v ,dx = int_Omega f v ,dx.

This is what is usually called the weak formulation of Poisson's equation; what's missing is the space V. Well, this a bit tricky and way beyond the scope of this article. The space must allow us to write down this equation. Therefore, we should require that the derivatives of functions in this space are square integrable. Now, there is actually the Sobolev space H^1_0(Omega) of functions with weak derivatives in L^2(Omega) and with zero boundary conditions, which fulfills this purpose.

We obtain the generic form by assigning

: a(u,v) = int_Omega abla u cdot abla v ,dx

and

: f(v) = int_Omega f v ,dx.

The Lax–Milgram theorem

This is a formulation of the Lax–Milgram theorem which relies on properties of the symmetric part of the bilinear form. It is not the most general form.

Let V be a Hilbert space and a(cdot,cdot) a bilinear form on V, which is

  1. bounded: |a(u,v)| le C |u| |v| and
  2. coercive: a(u,u) ge c |u|^2.

Then, for any fin V', there is a unique solution uin V to the equation

:a(u,v) = f(v)

and it holds

: |u| le frac1c |f|_{V'}.

Application to example 1

Here, application of the Lax–Milgram theorem is definitely overkill, but we still can use it and give this problem the same structure as the others have.

  • Boundedness: all bilinear forms on mathbb R^n are bounded. In particular, we have: |a(u,v)| le |A|,|u|,|v|
  • Coercivity: this actually means that the real parts of the eigenvalues of A are not smaller than c. Since this implies in particular that no eigenvalue is zero, the system is solvable.
Additionally, we get the estimate: |u| le frac1c |f|,where c is the minimal real part of an eigenvalue of A.

Application to Example 2

Here, as we mentioned above, we choose V = H^1_0(Omega) with the norm:|v|_V := | abla v|,

where the norm on the right is the L^2-norm on Omega.But, we see that a(u,u) = | abla u|^2 and by Cauchy-Schwarz inequality a(u,v) le | abla u|,| abla v|.

Therefore, for any fin [H^1_0(Omega)] ', there is a unique solution uin V of Poisson's equation and we have the estimate

:| abla u| le |f|_{ [H^1_0(Omega)] '}.

ee also

* Babuška-Lax-Milgram theorem

References

* cite book
last = Lax
first = Peter D.
authorlink = Peter Lax
coauthors = Milgram, Arthur N.
chapter = Parabolic equations
title = Contributions to the theory of partial differential equations
series = Annals of Mathematics Studies, no. 33
pages = 167–190
publisher = Princeton University Press
address = Princeton, N. J.
year = 1954
MathSciNet|id=0067317

External links

* [http://mathworld.wolfram.com/Lax-MilgramTheorem.html MathWorld page on Lax-Milgram theorem]


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