# Weak formulation

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{{ safesubst:#invoke:Unsubst||$N=Merge |date=__DATE__ |$B= Template:MboxTemplate:DMCTemplate:Merge partner }} 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 has instead weak solutions only with respect to certain "test vectors" or "test functions". This is equivalent to formulating the problem to require a solution in the sense of a distribution.

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

## General concept

$Au=f$ ,
$[Au](v)=f(v)$ .

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

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

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

$Au=f$ $\langle Au,v\rangle =\langle f,v\rangle ,\,$ Since $A$ is a linear mapping, it is sufficient to test with basis vectors, and we get

$\langle Au,e_{i}\rangle =\langle f,e_{i}\rangle \quad i=1,\ldots ,n.\,$ Actually, expanding $u=\sum _{j=1}^{n}u_{j}e_{j}$ , we obtain the matrix form of the equation

$\mathbf {A} \mathbf {u} =\mathbf {f} ,$ The bilinear form associated to this weak formulation is

$a(u,v)=\mathbf {v} ^{T}\mathbf {A} \mathbf {u} .$ ## Example 2: Poisson's equation

Our aim is to solve Poisson's equation

$-\nabla ^{2}u=f,\,$ $\langle u,v\rangle =\int _{\Omega }uv\,dx$ to derive our weak formulation. Then, testing with differentiable functions $v$ , we get

$-\int _{\Omega }(\nabla ^{2}u)v\,dx=\int _{\Omega }fv\,dx.$ We can make the left side of this equation more symmetric by integration by parts using Green's identity:

$\int _{\Omega }\nabla u\cdot \nabla v\,dx=\int _{\Omega }fv\,dx.$ This is what is usually called the weak formulation of Poisson's equation; what's missing is the space $V$ , which is 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 $H_{0}^{1}(\Omega )$ 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

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

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

$a(u,v)=f(v)$ and it holds

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

Additionally, we get the estimate

$\|u\|\leq {\frac {1}{c}}\|f\|,\,$ ### Application to example 2

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

$\|v\|_{V}:=\|\nabla v\|,$ $\|\nabla u\|\leq \|f\|_{[H_{0}^{1}(\Omega )]'}.$ 