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

Deprecated: MCO: to remove in favor of poisson_2d_3d_robin.md

This test case solves the 2D/3D Poisson equation with general Neumann boundary conditions on all boundaries.

The aims of this test case are to:

validate the discretization schemes used for the diffusion equation;
validate the boundary schemes using Neumann boundary conditions.

Results summary

Second-order is achieved with second-order schemes;
Fourth-order is achieved with fourth-order schemes.

Test case description

Configurations

The domain is the unit square/cube. We solve \(- \Delta T=f\) on the whole domain. Neumann boundary conditions are applied to all boundaries, where we set a uniform value \( \alpha \) (for homogeneous conditions, set \( \alpha = 0 \) ). Reference solutions of the problem are (see illustration below):

2D: \( T(x,y) = \cos(\pi x + \alpha)\ \cos(\pi y + \alpha)\ /\ \pi \)
3D: \( T(x,y,z) = \cos(\pi x + \alpha)\ \cos(\pi y + \alpha)\ \cos(\pi z + \alpha)\ /\ \pi \)

with the associated right-end-side:

2D: \( f(x,y) = 2 \pi \cos(\pi x + \alpha)\ \cos(\pi y + \alpha) \)
3D: \( f(x,y,z) = 3 \pi \cos(\pi x + \alpha)\ \cos(\pi y + \alpha)\ \cos(\pi z + \alpha) \)

Illustration

Runtime parameters

Energy equation is used;
Temporal and inertial terms are disabled;
Hypre solvers are used;
boundary_condition_scheme linear|quadratic|cubic;
diffusion_scheme implicit o2_centered|o4_centered;

Associated files

The convergence test files are: verification/laplacian/convergence/neumann_2D.nts and verification/laplacian/convergence/neumann_3D.nts.
The non-regression test file is: verification/laplacian/neumann.nts

Todo
IO: add these 2 convergence cases + keep only the verification/laplacian/convergence/neumann_3D.nts (=> neumann.nts) for non-regression for the 3 directions.

Results

2nd-order centered scheme with linear boundary scheme

Todo: IO: to be added (expected 1st order)

boundary_condition_scheme linear;

diffusion_scheme implicit o2_centered;

2nd-order centered scheme with quadratic boundary scheme

boundary_condition_scheme quadratic;

diffusion_scheme implicit o2_centered;

Second-order convergence is observed with \(L_2\), \(L_1\) or \(L_\infty\) norms.

2D mesh

mesh	\(L_\infty\) error	Order	\(L_1\) error	Order	\(L_2\) error	Order
10	2.566580e-03	n/a	1.075103e-03	n/a	1.315482e-03	n/a
20	6.512728e-04	1.98	2.661326e-04	2.01	3.276534e-04	2.01
40	1.634228e-04	1.99	6.636913e-05	2.00	8.183755e-05	2.00
80	4.089354e-05	2.00	1.658205e-05	2.00	2.045465e-05	2.00

3D mesh

mesh	\(L_\infty\) error	Order	\(L_1\) error	Order	\(L_2\) error	Order
10	2.534981e-03	n/a	6.872545e-04	n/a	9.301862e-04	n/a
20	6.492651e-04	1.97	1.695996e-04	2.02	2.316859e-04	2.01
40	1.632968e-04	1.99	4.226276e-05	2.00	5.786789e-05	2.00
80	4.088565e-05	2.00	1.055714e-05	2.00	1.446362e-05	2.00

4th-order centered scheme with cubic boundary scheme

boundary_condition_scheme quadratic;

diffusion_scheme implicit o2_centered;

The grid size starts from 10 in every direction to 160. Second-order convergence is observed with \(L_2\), \(L_1\) or \(L_\infty\) norms.

2D mesh

Mesh	Temperature L1 error	Order	Temperature L2 error	Order	Temperature Linf error	Order
10×10	1.1524e-05	n/a	1.4110e-05	n/a	3.0974e-05	n/a
20×20	8.0063e-07	3.85	9.9484e-07	3.83	2.1509e-06	3.85
40×40	5.2321e-08	3.94	6.5042e-08	3.94	1.3875e-07	3.95
80×80	3.3387e-09	3.97	4.1407e-09	3.97	8.6383e-09	4.01
160×160	2.1083e-10	3.99	2.6089e-10	3.99	5.3484e-10	4.01

3D mesh

Todo: IO: 3D mesh