Benchmark solution
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Calculation method
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We recall the system of equations that we solve:

:math:`\mathrm{\{}\begin{array}{cc}\mathrm{-}\mathrm{\nabla }\mathrm{\cdot }\sigma (u)+b\mathrm{\nabla }p& \text{=}f\\ {\mathrm{\partial }}_{t}(\mathrm{\nabla }\mathrm{\cdot }u)\mathrm{-}\kappa \Delta p& \text{=}0\end{array}`

where :math:`\sigma (u)\mathrm{=}\lambda \mathrm{\nabla }\mathrm{\cdot }u{I}_{d}+2\mu \varepsilon (u)` and :math:`{I}_{d}` refer to the identity matrix in dimension :math:`d`.

In 2D, we have the following analytical solution:

:math:`p(t,x,y)\mathrm{=}{e}^{\mathrm{-}\mathit{At}}\mathrm{sin}(\pi x)\mathrm{sin}(\pi y)`

:math:`u(t,x,y)\mathrm{=}\mathrm{-}\left[\begin{array}{c}\mathrm{cos}(\pi x)\mathrm{sin}(\pi y)\\ \mathrm{sin}(\pi x)\mathrm{cos}(\pi y)\end{array}\right]\frac{{e}^{\mathrm{-}\mathit{At}}}{2\pi }`

In 3D, we have the following analytical solution:

:math:`p(t,x,y)\mathrm{=}{e}^{\mathrm{-}\mathit{At}}\mathrm{sin}(\pi x)\mathrm{sin}(\pi y)\mathrm{sin}(\pi z)`

:math:`u(t,x,y)\mathrm{=}\mathrm{-}\left[\begin{array}{c}\mathrm{cos}(\pi x)\mathrm{sin}(\pi y)\mathrm{sin}(\pi z)\\ \mathrm{sin}(\pi x)\mathrm{cos}(\pi y)\mathrm{sin}(\pi z)\\ \mathrm{sin}(\pi x)\mathrm{sin}(\pi y)\mathrm{cos}(\pi z)\end{array}\right]\frac{{e}^{\mathrm{-}\mathit{At}}}{3\pi }`


Reference quantities and results
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The numerical solution is compared to the analytical solution at 3 distinct points of the mesh (in movements and in pressure).


Uncertainties about the solution
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Analytical solution