What are the conceptual differences in presenting a problem in strong or weak form? For example for a 2D Poisson problem the strong form is:
\begin{split}- \nabla^2 u(\pmb{x}) &= f(\pmb{x}),\quad \pmb{x}\mbox{ in } \Omega,
\\
u(\pmb{x}) &= u_0(\pmb{x}),\quad \pmb{x}\mbox{ on } \partial \Omega\thinspace .\end{split}
where $\Omega$ is the spatial domain and $\partial\Omega$ is the boundary of $\Omega$.
The variational or weak formulation:
\begin{equation} \int_{\Omega} \nabla u \cdot \nabla v \, \mathrm{d}x =
\int_{\Omega} fv \, \mathrm{d}x
\quad \forall v \in \hat{V}.\
\end{equation}
where $\hat{V}$ is the test space and $V$ is the trial space:
\begin{split}\hat{V} &= \{v \in H^1(\Omega) : v = 0 \mbox{ on } \partial\Omega\}, \\
V &= \{v \in H^1(\Omega) : v = u_0 \mbox{ on } \partial\Omega\}\thinspace .\end{split}
I know that the weak form is very useful in the Finite Element Method, but I don't understand why. The wikipedia page says that the problem requieres a solution in the sense of a distribution. What does this mean?
Why are they called Strong and Weak? What is the intuition behind this formulations?
Thanks!
Best Answer
If we say a solution is weak/strong/classical/viscous, the following aspects are concerned (or more):
How we obtain the solution.
The regularity of the solution (how smooth this solution is, integrability, differentiability).
The solution satisfies the equation in what sense.
Weak solution:
Smoothness depends on the right side data. If the $f\in H^{-1}$, then $u\in H^1$. If $f\in L^2$, then $u\in H^2_{loc}$. Moreover if $\Omega$ is $C^{1,1}$, we have an $H^2$-solution $u$ globally.
The solution satisfies the equation in distribution sense (see following explanation).
Why "weak":
The term "weak" normally refers to the 2 and 3: The solution $u$ is only in $H^1$ in the most general setting, this means that $u$ is the only differentiable once, notice $-\Delta$ has second partial derivative in it. The strong solution, however, indeed have twice differentiability, normally if we say $u$ is a strong solution, we mean that $u$ has $W^{2,p}$-regularity (Please refer to Gilbarg and Trudinger). The solution satisfies the equation only in the "weak" formulation $$ \int_{\Omega} \nabla u \cdot \nabla v \, dx = \int_{\Omega} fv \, dx \quad \forall v \in V, \tag{1} $$ where $V$ is certain Sobolev space.
Two ways to get this weak form: first is to write what condition the minimizer of $\mathcal{F}(u)$ must satisfy: if $u$ is a minimizer, then $$ \lim_{\epsilon \to 0}\frac{d}{d\epsilon} \mathcal{F}(u+\epsilon v) =0 $$ and the weak form of Euler-Lagrange equation is (1).
Another is multiplying the original equation by a test function then integration by parts. The intuition behind this should be Riesz representation theorem (at least to me it makes sense), we have: $$ \langle (-\Delta)u,v\rangle = l_u(v) = (u,v)_{V}, $$ from the differential operator $-\Delta$ $\to$ linear functional $l_u$ $\to$ representation using inner product $(\cdot ,\cdot)_V$. The inner product $(\cdot ,\cdot)_V$ on this Hilbert space $V$ is the left hand side of (1), if we make the test function space have zero boundary condition (We can use Poincaré inequality to prove the equivalence of the standard $H^1$-inner product). If you have taken any numerical PDE course in finite element, the professor would introduce Lax-Milgram theorem, and Lax-Milgram relies on Riesz.
Why weak form is useful in finite element method:
Short answer: Weak form is very handy in that it helps us formulate a linear equation system which can be solved by computer!
Long answer: The essential of Galerkin type approach is that we are exploiting the fact that the infinite dimensional Hilbert space has a set of basis $\{\phi_i\}_{i=1}^{\infty}$, if we can expand the $u$ in this basis: $$ u = \sum_{i=1}^{\infty} u_i\phi_i, $$ where $u_n$ is a number, pluggin back to (1), and let the test function $v$ run through all $\phi_j$ (same function, different subscript): $$ \int_{\Omega} \nabla (\sum_{i=1}^{\infty} u_i\phi_i) \cdot \nabla \phi_j \, dx =\sum_{i=1}^{\infty} u_i \int_{\Omega} \nabla \phi_i \cdot \nabla \phi_j \, dx = \int_{\Omega} f\phi_j \, dx \quad \forall j =1,2,\ldots. \tag{2} $$ We have an infinite dimensional linear equation system: $$ AU = F, $$ where $A_{ji} = \displaystyle\int_{\Omega} \nabla \phi_i \cdot \nabla \phi_j \, dx$, $U_i = u_i$, and $F_j = \displaystyle\int_{\Omega} f\phi_j\, dx$.
Finite element method essentially choose a finite dimensional subspace $V_h\subset V$ (may not be a subspace, please google Discontinuous Galerkin method), so that we approximate the solution in this finite dimensional subspace $V_h$! The summation in (2) does not have an infinite upper limit any more, instead there are finitely many $\phi_i$ and $v$ runs from $\phi_1$ to $\phi_N$, so that the linear system generated is still $AU = F$, but this time, it only has $N$ equations, and we can use computer to solve it.