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Wikipedia / Weak formulation

Weak formulations are 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

be a Banach space be the dual space be a linear map is a solution of the equation

,

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

such that

by defining the bilinear form

Example 1: linear system of equations

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

the following equation holds:

denotes an inner product.

Since the inner product is bilinear, it is sufficient to test with basis vectors, and we get

Actually, expanding we obtain the matrix form of the equation

and

The bilinear form associated to this weak formulation is

Example 2: Poisson's equation

To solve Poisson's equation

on its boundary later, one can use the scalar product

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

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

This is what is usually called the weak formulation of Poisson's equation. Functions 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 and with zero boundary conditions, so

The generic form is obtained by assigning

and

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.

be a real Hilbert space a bilinear form on which is

  1. bounded and

  2. coercive

Then, for any bounded to the equation

and it holds

Application to example 1

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

  • Coercivity: this actually means that the real parts of the eigenvalues. Since this implies in particular that no eigenvalue is zero, the system is solvable.

is the minimal real part of an eigenvalue of

Application to example 2

where the norm on the right is the by the Poincaré inequality and by the Cauchy–Schwarz inequality,

Therefore, for any of Poisson's equation and we have the estimate

See also

References