Weak formulation

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.

In this article weak formulations are introduced by a few examples, and then the main theorem for the solution, the Lax–Milgram theorem, is presented. The theorem is named after Peter Lax and Arthur Milgram, who proved it in 1954.

General concept[]

Let ${\displaystyle V}$ be a Banach space. We want to find the solution ${\displaystyle u\in V}$ of the equation

${\displaystyle Au=f,}$

where ${\displaystyle A\colon V\to V'}$ and ${\displaystyle f\in V'}$, with ${\displaystyle V'}$ being the dual space of ${\displaystyle V}$.

This is equivalent to finding ${\displaystyle u\in V}$ such that for all ${\displaystyle v\in V}$,

${\displaystyle [Au](v)=f(v).}$

Here, we call ${\displaystyle v}$ a test vector or test function.

We bring this into the generic form of a weak formulation, namely, find ${\displaystyle u\in V}$ such that

${\displaystyle a(u,v)=f(v)\quad \forall v\in V,}$

by defining the bilinear form

${\displaystyle a(u,v):=[Au](v).}$

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

Example 1: linear system of equations[]

Now, let ${\displaystyle V=\mathbb {R} ^{n}}$ and ${\displaystyle A:V\to V}$ be a linear mapping. Then, the weak formulation of the equation

${\displaystyle Au=f}$

involves finding ${\displaystyle u\in V}$ such that for all ${\displaystyle v\in V}$ the following equation holds:

${\displaystyle \langle Au,v\rangle =\langle f,v\rangle ,}$

where ${\displaystyle \langle \cdot ,\cdot \rangle }$ denotes an inner product.

Since ${\displaystyle A}$ is a linear mapping, it is sufficient to test with basis vectors, and we get

${\displaystyle \langle Au,e_{i}\rangle =\langle f,e_{i}\rangle ,\quad i=1,\ldots ,n.}$

Actually, expanding ${\displaystyle u=\sum _{j=1}^{n}u_{j}e_{j}}$, we obtain the matrix form of the equation

${\displaystyle \mathbf {A} \mathbf {u} =\mathbf {f} ,}$

where ${\displaystyle a_{ij}=\langle Ae_{j},e_{i}\rangle }$ and ${\displaystyle f_{i}=\langle f,e_{i}\rangle }$.

The bilinear form associated to this weak formulation is

${\displaystyle a(u,v)=\mathbf {v} ^{T}\mathbf {A} \mathbf {u} .}$

Example 2: Poisson's equation[]

Our aim is to solve Poisson's equation

${\displaystyle -\nabla ^{2}u=f,}$

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

${\displaystyle \langle u,v\rangle =\int _{\Omega }uv\,dx}$

to derive our weak formulation. Then, testing with differentiable functions ${\displaystyle v}$, we get

${\displaystyle -\int _{\Omega }(\nabla ^{2}u)v\,dx=\int _{\Omega }fv\,dx.}$

We can make the left side of this equation more symmetric by integration by parts using Green's identity and assuming that ${\displaystyle v=0}$ on ${\displaystyle \partial \Omega }$:

${\displaystyle \int _{\Omega }\nabla u\cdot \nabla v\,dx=\int _{\Omega }fv\,dx.}$

This is what is usually called the weak formulation of Poisson's equation. We have yet to specify a space ${\displaystyle V}$ in which to find a solution, but at a minimum it must allow us to write down this equation. Therefore, we require that the functions in ${\displaystyle V}$ are zero on the boundary, and have square-integrable derivatives. The appropriate space to satisfy these requirements is the Sobolev space ${\displaystyle H_{0}^{1}(\Omega )}$ of functions with weak derivatives in ${\displaystyle L^{2}(\Omega )}$ and with zero boundary conditions, so we set ${\displaystyle V=H_{0}^{1}(\Omega )}$.

We obtain the generic form by assigning

${\displaystyle a(u,v)=\int _{\Omega }\nabla u\cdot \nabla v\,dx}$

and

${\displaystyle f(v)=\int _{\Omega }fv\,dx.}$

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 ${\displaystyle V}$ be a Hilbert space and ${\displaystyle a(\cdot ,\cdot )}$ a bilinear form on ${\displaystyle V}$, which is

1. bounded: ${\displaystyle |a(u,v)|\leq C\|u\|\|v\|}$ and
2. coercive: ${\displaystyle a(u,u)\geq c\|u\|^{2}.}$

Then, for any ${\displaystyle f\in V'}$, there is a unique solution ${\displaystyle u\in V}$ to the equation

${\displaystyle a(u,v)=f(v)\quad \forall v\in V}$

and it holds

${\displaystyle \|u\|\leq {\frac {1}{c}}\|f\|_{V'}.}$

Application to example 1[]

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

• Boundedness: all bilinear forms on ${\displaystyle \mathbb {R} ^{n}}$ are bounded. In particular, we have
  $${\displaystyle |a(u,v)|\leq \|A\|\,\|u\|\,\|v\|}$$

  
• Coercivity: this actually means that the real parts of the eigenvalues of ${\displaystyle A}$ are not smaller than ${\displaystyle c}$. Since this implies in particular that no eigenvalue is zero, the system is solvable.

Additionally, we get the estimate

where ${\displaystyle c}$ is the minimal real part of an eigenvalue of ${\displaystyle A}$.

Application to example 2[]

Here, as we mentioned above, we choose ${\displaystyle V=H_{0}^{1}(\Omega )}$ with the norm

${\displaystyle \|v\|_{V}:=\|\nabla v\|,}$

where the norm on the right is the ${\displaystyle L^{2}}$-norm on ${\displaystyle \Omega }$ (this provides a true norm on ${\displaystyle V}$ by the Poincaré inequality). But, we see that ${\displaystyle |a(u,u)|=\|\nabla u\|^{2}}$ and by the Cauchy–Schwarz inequality, ${\displaystyle |a(u,v)|\leq \|\nabla u\|\,\|\nabla v\|}$.

Therefore, for any ${\displaystyle f\in [H_{0}^{1}(\Omega )]'}$, there is a unique solution ${\displaystyle u\in V}$ of Poisson's equation and we have the estimate

${\displaystyle \|\nabla u\|\leq \|f\|_{[H_{0}^{1}(\Omega )]'}.}$

See also[]

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

References[]

• Lax, Peter D.; Milgram, Arthur N. (1954), "Parabolic equations", Contributions to the theory of partial differential equations, Annals of Mathematics Studies, vol. 33, Princeton, N. J.: Princeton University Press, pp. 167–190, doi:10.1515/9781400882182-010, MR 0067317, Zbl 0058.08703

External links[]

• MathWorld page on Lax–Milgram theorem