# Split of complex parts in weak form

I am working on a numerical model to simulate the acoustic and elastic wave propagation in frequency domain via the Finite Element Method. Basically, the problem is to solve the Helmholtz equation in the fluid domain, the Navier-Cauchy equation in the solid domain and a couple of interface conditions.

More precisely, these interface conditions are the continuity of the normal component of the velocity and the continuity of traction, both on the interface $$\Gamma$$:

$$\cases{ \partial_{\mathbf{n}}p = \rho_F \omega^2 \mathbf{n} \cdot \mathbf{u} \\ -p \mathbf{n} = \sigma(\mathbf{u}) \cdot \mathbf{n} }$$

Here, $$p$$ is the deviation from ambient pressure, $$\sigma$$ the Cauchy stress tensor, $$\rho_F$$ fluid density, $$\omega$$ the angular frequency, $$\mathbf{u}$$ the displacement vector in $$\Omega_S$$ and $$\mathbf{n}$$ the unit normal vector of $$\Gamma$$. Their weak form are respectevely:

$$\int_{\Gamma} \rho_F \omega^2 (\mathbf{n} \cdot \mathbf{u}) q ds = 0 \quad (1)$$ and $$\int_{\Gamma} (p \mathbf{n}) \cdot \mathbf{v} ds = 0 \quad (2)$$

where $$q$$ and $$\mathbf{v}$$ are test functions. Now I want to split $$p$$, $$\mathbf{u}$$, $$q$$ and $$\mathbf{v}$$ into its real and imaginary parts. I simply did $$p = p^R + ip^I$$, $$\mathbf{u} = \mathbf{u}^R + i\mathbf{u}^I$$, $$q = q^R + iq^I$$ and $$\mathbf{v} = \mathbf{v}^R + i\mathbf{v}^I$$. By apllying the distributive property, I rewrote $$(1)$$ and $$(2)$$, respectively, as

$$\int_{\Gamma} \rho_F \omega^2 [\mathbf{n} \cdot \mathbf{u}^R q^R - \mathbf{n} \cdot \mathbf{u}^I q^I + i(\mathbf{n} \cdot \mathbf{u}^R q^I + \mathbf{n} \cdot \mathbf{u}^I q^R)] ds = 0$$

and

$$\int_{\Gamma} [(p^R \mathbf{n}) \cdot \mathbf{v}^R - (p^I \mathbf{n}) \cdot \mathbf{v}^I + i((p^R \mathbf{n}) \cdot \mathbf{v}^I - (p^I \mathbf{n}) \cdot \mathbf{v}^R)] ds = 0$$

Well, since I have little experience on this topic, I am afraid it is not as simple as that. Can someone confirm it or correct me if I am wrong?

• Like for most other things, there is a deal.II tutorial program about the issue: dealii.org/developer/doxygen/deal.II/step_29.html – Wolfgang Bangerth Sep 21 '20 at 3:53
• Thank you for your answer @WolfgangBangerth. I see that $v$ and $w$ represent respectively the Real and Imaginary part of $u$. However, I'm not sure if $\phi$ and $\psi$ are also the Real and the Imaginary parts of the test function. If that's the case, why do we have $\langle \phi, w \rangle$ for the Real part and $\langle \psi, v \rangle$ for the Imaginary of the weak form ? – Lucas Vieira Sep 26 '20 at 18:55
• $\phi$ denotes the test function for the first equation (the real part of the original equation) and $\psi$ denotes the test function for the second equation. Both are real-values, but they do not correspond to real or imaginary parts of a hypothesized complex-valued test function. – Wolfgang Bangerth Sep 28 '20 at 4:03