2. Benchmark solution

2.1. Calculation method

We recall the system of equations that we solve:

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

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

In 2D, we have the following analytical solution:

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

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

In 3D, we have the following analytical solution:

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

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

2.2. Reference quantities and results

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

2.3. Uncertainties about the solution

Analytical solution