Weak formulation

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 of ${\displaystyle V}$.

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

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

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.

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

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

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

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

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

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

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

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

• MathWorld page on Lax–Milgram theorem