Weak formulation explained

Weak formulations are important tools 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, equations or conditions are no longer required to hold absolutely (and this is not even well defined) and has instead weak solutions only with respect to certain "test vectors" or "test functions". In a strong formulation, the solution space is constructed such that these equations or conditions are already fulfilled.

The Lax–Milgram theorem, named after Peter Lax and Arthur Milgram who proved it in 1954, provides weak formulations for certain systems on Hilbert spaces.

General concept

Let

be a Banach space, let be the dual space of , let , and let . A vector is a solution of the equation

$$Au\; =\; f$$

if and only if for all

,

$$(Au)(v)\; =\; f(v).$$

A particular choice of

is called a test vector (in general) or a test function (if is a function space).

To bring this into the generic form of a weak formulation, find

such that

$$a(u,v)\; =\; f(v)\; \backslash quad\; \backslash forall\; v\; \backslash in\; V,$$

by defining the bilinear form

$$a(u,v)\; :=\; (Au)(v).$$

Example 1: linear system of equations

Now, let

and be a linear mapping. Then, the weak formulation of the equation

$$Au\; =\; f$$

involves finding

such that for all the following equation holds:

$$\backslash langle\; Au,v\; \backslash rangle\; =\; \backslash langle\; f,v\; \backslash rangle,$$

where

denotes an inner product.

Since

is a linear mapping, it is sufficient to test with basis vectors, and we get

$$\backslash langle\; Au,e\_i\backslash rangle\; =\; \backslash langle\; f,e\_i\backslash rangle,\; \backslash quad\; i=1,\backslash ldots,n.$$

Actually, expanding we obtain the matrix form of the equation

$$\backslash mathbf\backslash mathbf\; =\; \backslash mathbf,$$

where

and

The bilinear form associated to this weak formulation is

$$a(u,v)\; =\; \backslash mathbf^T\backslash mathbf\; \backslash mathbf.$$

Example 2: Poisson's equation

$$-\backslash nabla^2\; u\; =\; f,$$

on a domain

with on its boundary, and to specify the solution space later, one can use the scalar product

$$\backslash langle\; u,v\backslash rangle\; =\; \backslash int\_\backslash Omega\; uv\backslash ,dx$$

to derive the weak formulation. Then, testing with differentiable functions yields

$$-\backslash int\_\backslash Omega\; (\backslash nabla^2\; u)\; v\; \backslash ,dx\; =\; \backslash int\_\backslash Omega\; fv\; \backslash ,dx.$$

The left side of this equation can be made more symmetric by integration by parts using Green's identity and assuming that

on

$$\backslash int\_\backslash Omega\; \backslash nabla\; u\; \backslash cdot\; \backslash nabla\; v\; \backslash ,dx\; =\; \backslash int\_\backslash Omega\; f\; v\; \backslash ,dx.$$

This is what is usually called the weak formulation of Poisson's equation. Functions in the solution space

must be zero on the boundary, and have square-integrable derivatives. The appropriate space to satisfy these requirements is the Sobolev space of functions with weak derivatives in and with zero boundary conditions, so

The generic form is obtained by assigning

$$a(u,v)\; =\; \backslash int\_\backslash Omega\; \backslash nabla\; u\; \backslash cdot\; \backslash nabla\; v\; \backslash ,dx$$

and

$$f(v)\; =\; \backslash int\_\backslash Omega\; f\; v\; \backslash ,dx.$$

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

be a Hilbert space and a bilinear form on which is

1. bounded:

and

1. coercive:

Then, for any bounded there is a unique solution

to the equation

$$a(u,v)\; =\; f(v)\; \backslash quad\; \backslash forall\; v\; \backslash in\; V$$

and it holds

$$\backslash |u\backslash |\; \backslash le\; \backslash frac1c\; \backslash |f\backslash |\_\backslash ,.$$

Application to example 1

Here, application of the Lax–Milgram theorem is a stronger result than is needed.

• Boundedness: all bilinear forms on

are bounded. In particular, we have $$|a(u,v)|\; \backslash le\; \backslash |A\backslash |\backslash ,\backslash |u\backslash |\backslash ,\backslash |v\backslash |$$

• Coercivity: this actually means that the real parts of the eigenvalues of

are not smaller than . Since this implies in particular that no eigenvalue is zero, the system is solvable.

Additionally, this yields the estimate$$\backslash |u\backslash |\; \backslash le\; \backslash frac1c\; \backslash |f\backslash |,$$where

is the minimal real part of an eigenvalue of

Application to example 2

Here, choose

with the norm$$\backslash |v\backslash |\_V\; :=\; \backslash |\backslash nabla\; v\backslash |,$$

where the norm on the right is the norm on

(this provides a true norm on by the Poincaré inequality).But, we see that and by the Cauchy–Schwarz inequality,

Therefore, for any there is a unique solution

of Poisson's equation and we have the estimate

$$\backslash |\backslash nabla\; u\backslash |\; \backslash le\; \backslash |f\backslash |\_.$$

See also

• Babuška–Lax–Milgram theorem
• Lions–Lax–Milgram theorem

External links

• MathWorld page on Lax–Milgram theorem