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Update projectPBE.md
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docs/_projects/projectPBE.md

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-\Delta u + k^2 \sinh (u) = \rho (x), \quad x\in \Omega\subset \mathbb R^d,
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with Dirichlet boundary condition $u|_{\partial \Omega} = g.$​​
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We consider a 2-d problem on the unit square $\Omega = (0,1)^2$. Let $\boldsymbol a=(1.0,2.0)/\sqrt{5}$. We choose $k =1, \rho$, and $g$ such that the exact solution is
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We consider a 2-d problem on the unit square $\Omega = (0,1)^2$. Let $\boldsymbol a=(1.0,2.0)/\sqrt{5}$. We choose $k =1, \rho$, and $g$ such that the exact solution is
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$$
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u(\boldsymbol x) = \bar u(0.1+\boldsymbol a\cdot\boldsymbol x),\quad \bar u(s) = \ln \left ( \frac{1+\cos (s)}{1-\cos (s)}\right).

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