2D/3D Poisson test cases with Robin boundary conditions

This test case solves the 2D/3D Poisson equation with Robin boundary conditions using a second and fourth-order centered scheme and the fourth-order boundary scheme cubic. The aims of this test case are:

validate the discretization of the diffusive term of a cell advection/diffusion equation;
validate the Robin boundary condition of a cell scalar equation in higher dimensions;
validate all the schemes i higher dimensions.

Configurations

Setup

Domain is cubic, its opposite corner coordinates being \((0,0,0)\) and \((1,1,1)\). We solve \(- \Delta T=f\) on the whole domain.

2D solution of the problem is set to

\[ T(x,y) = \cos(\pi x) \cos(\pi y)+ \sin(\pi x) \sin(\pi y)\]

.

3D solution of the problem is set to

\[ T(x,y,z) = \cos(\pi x) \cos(\pi y) \cos(\pi z)+ \sin(\pi x) \sin(\pi y) \sin(\pi z)\]

.

The right-end-side term \( f \) is equal to

In 2D : \( f(x) = 2\pi^2 (\cos(\pi x) \cos(\pi y)+ \sin(\pi x) \sin(\pi y))\).
In 3D : \( f(x) = 3\pi^2 (\cos(\pi x) \cos(\pi y) \cos(\pi z)+ \sin(\pi x) \sin(\pi y) \sin(\pi z))\).

The Robin boundary condition :

\[ T(x) = \alpha T(x) + \beta \nabla T . \vec{n} = g \]

.

Where \( \alpha = (\alpha_x ,\alpha_y, \alpha_z) \) and \( \beta = ( \beta_x , \beta_y, \beta_z) \) are the Robin coefficient in each direction and g is the value boundary condition and lastly \( \vec{n} \) is the normal vector on the boundary

Robin parameteres in 3D are chosen as follows.

Boundary	\(\alpha \)	\(\beta \)	\(g \)
Left	1	10	\( \alpha_x \pi \cos(\pi y ) \cos(\pi z ) + \beta_x \pi \sin(\pi y ) \sin(\pi z )\)
Right	10	5	\( -\alpha_x \pi \cos(\pi y ) \cos(\pi z ) - \beta_x \pi \sin(\pi y ) \sin(\pi z ) \)
bottom	1	10	\( \alpha_y \pi \cos(\pi x ) \cos(\pi z ) + \beta_y \pi \sin(\pi x ) \sin(\pi z )\)
top	10	5	\( -\alpha_y \pi \cos(\pi x ) \cos(\pi z ) - \beta_y \pi \sin(\pi x ) \sin(\pi z ) \)

Robin parameteres in 3D are chosen as follows.

Boundary	\(\alpha \)	\(\beta \)	\(g \)
Left	1	10	\( \alpha_x \pi \cos(\pi y ) \cos(\pi z ) + \beta_x \pi \sin(\pi y ) \sin(\pi z )\)
Right	10	5	\( -\alpha_x \pi \cos(\pi y ) \cos(\pi z ) - \beta_x \pi \sin(\pi y ) \sin(\pi z ) \)
bottom	1	10	\( \alpha_y \pi \cos(\pi x ) \cos(\pi z ) + \beta_y \pi \sin(\pi x ) \sin(\pi z )\)
top	10	5	\( -\alpha_y \pi \cos(\pi x ) \cos(\pi z ) - \beta_y \pi \sin(\pi x ) \sin(\pi z ) \)
back	1	10	\( \alpha_z \pi \cos(\pi y ) \cos(\pi x ) + \beta_z \pi \sin(\pi y ) \sin(\pi x )\)
front	10	5	\( -\alpha_z \pi \cos(\pi y ) \cos(\pi x ) - \beta_z \pi \sin(\pi y ) \sin(\pi x ) \)

Runtime parameters

Energy equation is used:

Temporal and inertial terms are disabled;
HYPRE solver is used;
Both the fourth or second-order for diffusion scheme can be used.
Only cubic schemes for the boundary conditions is used.

Test cases

The file test_cases/verification/laplacian/bc_schemes_2D_3D.nts is used to check the convergence as well as to verify the non-regression.

Comments and Expectations

Since the analytical solution is infinitely smooth (regular), we expect the numerical schemes to reproduce the solution with very high accuracy.

For the second-order centered scheme, the convergence rate should clearly approach O(h²) once the mesh is sufficiently refined.
For the fourth-order centered scheme, we expect an even faster decay of the error, following O(h⁴).

Regarding the Robin boundary conditions:

With cubic boundary schemes, the boundary approximation matches the smoothness of the exact solution, allowing the scheme to recover the expected order of accuracy of the interior discretization.

In summary, even in higher dimensions because the solution is highly regular, the numerical error should decrease rapidly with mesh refinement, and the impact of the boundary condition discretization order will be clearly visible in the convergence rates.

Results

Second-order diffusion scheme for all boundary schemes

boundary_condition_scheme linear | quadratic | cubic;

diffusion_scheme implicit o2_centered;

A convergence study is regrouped in following graph. Second order convergence is observed with \(L_2\), \(L_1\) or \(L_\infty\) norms with all the boudary schemes.

Figure 1: Convergence graph with multiple boundary schemes

Fourth-order diffusion scheme for all boundary schemes

boundary_condition_scheme linear | quadratic | cubic;

diffusion_scheme implicit o4_centered;

A convergence study is regrouped in following graph. The lowest order between the inner scheme and boundary scheme convergence is observed with \(L_2\), \(L_1\) or \(L_\infty\) norms with all the boundary schemes.

Figure 2: Convergence graph with multiple boundary schemes