# Deriving elliptic PDE from the weak form

Let $$\Omega \subset \mathbb R^n$$ and $$\Gamma \subset \partial \Omega$$ and $$\Gamma_n=\partial \Omega \setminus \Gamma$$, $$V_0=\{v \in H^1(\Omega) : v=0 \text{ on } \Gamma\}$$.

Given the variational statements $$a(u,v)=\int_\Omega a(x)\nabla u(x) \cdot \nabla v(x)+\bigl (b(x) \cdot \nabla u(x) \bigr)v(x) \, \mathrm{dx}$$ or $$a(u,v)=\int_\Omega a(x)\nabla u(x) \cdot \nabla v(x)+\bigl [\nabla \cdot (b(x) v(x)) \bigr]u(x) \, \mathrm{dx}$$ for $$b\in \mathbb R^n$$ where we seek $$u\in V_0$$ such that $$a(u,v)=f(v):=\int_{\Omega} f(x)v(x)\, \mathrm{dx}+ \int_{\Gamma} g(x)v(x)\, \mathrm{ds}$$ for some $$f \in L^2(\Omega)$$ and all $$v \in V_0$$, how do I find the corresponding elliptic PDE and boundary conditions asssuming extra regularity? For "simple" weak forms such as $$a(u,v)=\int_\Omega a(x)\nabla u(x) \cdot \nabla v(x)+c(x)u(x)v(x) \, \mathrm{dx}$$ it is just $$-a(x)\Delta u(x)+c(x)u(x) =f(x)$$ in $$\Omega$$ and $$u=0$$ on $$\partial \Omega$$ if I got this right.

• Do the terms with $a$ really contain only $v$ or should it be $\nabla v$?
– gerw
Commented Jan 17, 2020 at 11:39
• @gerw thanks, forgot it and then copy pasted it into other integrals Commented Jan 17, 2020 at 12:06
• @Tesla How do you define $f(v)$? Commented Jan 18, 2020 at 12:48
• @ahdahmanii sorry added details to it Commented Jan 18, 2020 at 13:19

Assuming that $$\Omega$$ is Lipschitz- continuous subset of $$\mathbb{R}^n$$ and making use of the identity I.2.17 and theorem 2.4 and 2.5 in [V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, 5 Springer, Berlin, 1986.] Which state that\

• $$\mathcal{D}(\bar{\Omega})^n$$ is dense in $$H(\operatorname{div},\Omega)$$\

and

• the map $$\gamma: v \rightarrow v\cdot\eta|_{\Gamma_n}$$ defined on $$\mathcal{D}(\bar{\Omega})^n$$ can be extended by continuity to a linear and continuous mapping from $$H(\operatorname{div},\Omega)$$ into $$H^{-\frac{1}{2}}(\Gamma_n)$$\ So, we have the following Green formula

$$(v,\nabla \phi)+(\operatorname{div} v,\phi)=(v\cdot \eta,\phi)_{\Gamma_n} \quad \forall v \in H(\operatorname{div},\Omega), \forall \phi \in H^1(\Omega)$$

Therefore, we have for every $$v \in V_0$$

\begin{align} a(u,v)=\int_\Omega a(x)\nabla u(x) \cdot \nabla v(x)+\bigl (b(x) \cdot \nabla u(x) \bigr)v(x) \, \mathrm{dx} \\ = \int_\Omega -\operatorname{div} \big(a(x)\nabla u(x)\big)v(x)+\bigl (b(x) \cdot \nabla u(x) \bigr)v(x) \, \mathrm{dx}\\+\int_{\Gamma_n} a(x)\nabla u(x) \eta v(x) \mathrm{ds} \end{align}

while $$\eta$$ is the outward unit normal vector. Now, taking $$v \in \mathcal{D}(\Omega)$$ we have $$\int_\Omega -\operatorname{div} \big(a(x)\nabla u(x)\big)v(x)+\bigl (b(x) \cdot \nabla u(x) \bigr)v(x) \, \mathrm{dx}=\int_{\Omega} f(x)v(x)\, \mathrm{dx}$$ So, $$-\operatorname{div} \big(a(x)\nabla u(x)\big)+\bigl (b(x) \cdot \nabla u(x) \bigr)=f(x) \quad \text{a.e. in } \Omega$$ Which can be extended into all $$\Omega$$

Making use of this, we find $$a(v) \frac{\partial u}{\partial \eta}=g(x), \quad \forall x \in \Gamma_n$$

Finely, the equation is

\left\{\begin{aligned} -\operatorname{div} \big(a(x)\nabla u(x)\big)+\bigl (b(x) \cdot \nabla u(x) \bigr)=f(x) \quad \forall x \in \Omega\\ a(x) \frac{\partial u}{\partial \eta}=g(x), \quad \forall x \in \Gamma_n \end{aligned}\right.

The second one can be treated the same way to get

\left\{\begin{aligned} -\operatorname{div} \big(a(x)\nabla u(x)\big)-\bigl (b(x) \cdot \nabla u(x) \bigr)=f(x) \quad \forall x \in \Omega\\ a(x) \frac{\partial u}{\partial \eta}+b(x)u(x)\eta=g(x), \quad \forall x \in \Gamma_n \end{aligned}\right.

• Thanks a lot, will have a look at the identities you used. Do you happen to know the equation to the second weak form I stated? Commented Jan 18, 2020 at 14:24
• And do the identies 2.17 and theorem 2.3, 2.5 have names so I can look them up on the internet? Commented Jan 18, 2020 at 14:26
• @Tesla, yes, I'm sorry, I have fixed it Commented Jan 18, 2020 at 15:27
• thanks a lot. can you just state the theorems very shortly that you used so that i can understand the first steps? Commented Jan 18, 2020 at 15:29
• @Tesla further details added. look at it now Commented Jan 18, 2020 at 15:58