I have two dimensional equation and I want to solve it using Finite Element Methods. $$ \nabla . (\alpha(x,y)\nabla u(x,y)) + \dfrac{\partial u(x,y)}{\partial x}+\dfrac{\partial u(x,y)}{\partial y}+u(x,y)=f(x,y)$$ $$\alpha(x,y)\dfrac{\partial u(x,y)}{\partial n}=g(x,y), for (x,y)\in \partial\Omega$$ To obtain its weak form, I multiplied the equation with a residual function $w$. And applied the vector identity shown below. $$\nabla.(b\vec{A})=\nabla b.\vec{A}+b\nabla . \vec{A}$$ One of the terms is come out as shown below. $$\int_{\partial\Omega} w\nabla g(x,y).\vec{n} dl $$ This term corresponds to a boundary condition. How should I derive the rest of the equation?


Dont expand the divergence term, apply integration by parts without doing this $$ \int_\Omega w \nabla\cdot(\alpha\nabla u) dx = - \int_\Omega \alpha \nabla u \cdot \nabla w dx + \int_{\partial\Omega} w \alpha \frac{\partial u}{\partial n}ds $$ In your notation, define a vector field $$ \vec{A} = \alpha \nabla u $$ and then do the integration by parts on $$ \int_\Omega w \nabla\cdot\vec{A} dx $$ Since $$ w \nabla\cdot\vec{A} = \nabla\cdot(w\vec{A}) - \vec{A} \cdot \nabla w $$ You get $$ \int_\Omega w \nabla\cdot\vec{A} dx = \int_{\partial\Omega} w \vec{A}\cdot\vec{n} ds - \int_\Omega \vec{A} \cdot \nabla w dx $$ which is the first equation above. Using your boundary condition, it becomes $$ \int_\Omega w \nabla\cdot(\alpha\nabla u) dx = - \int_\Omega \alpha \nabla u \cdot \nabla w dx + \int_{\partial\Omega} w g ds $$

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  • $\begingroup$ Shouldn't the first equation be $ \int_\Omega w \nabla\cdot(\alpha\nabla u) ds = - \int_\Omega \alpha \nabla u \cdot \nabla w ds + \int_{\partial\Omega} w \alpha \frac{\partial u}{\partial n}dl $ ? @cfdlab $\endgroup$ – Aldrich Taylor Apr 30 at 19:16
  • $\begingroup$ On the other hand, how did you find the $\int_{\partial\Omega} w \vec{A}\cdot\vec{n} ds$ term? @cfdlab $\endgroup$ – Aldrich Taylor Apr 30 at 22:36
  • $\begingroup$ It is divergence theorem en.wikipedia.org/wiki/Divergence_theorem. Replace $ds=dl$ to get your notation. I used $dx$ for domain integration and $ds$ for surface integration. $\endgroup$ – cfdlab May 1 at 3:21

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