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# Dumbbell shaped region (starting from smaller cavity)

## Introduction

This region has two connected cavities, with a ball inside larger cavity. Wave function stays in a smaller cavity.

## Equation to solve

 i ∂Ψ(x,y,z) ∂t + ∆Ψ(x,y,z) + |Ψ(x,y,z)|2Ψ(x,y,z) − 0.3 |Ψ(x,y,z)|4Ψ(x,y,z) = 0

## Domain

The region is bounded by a rectangular box [−1,1]×[−1.5,3]×[−1,1]. Large cavity is a ball of radius 1 with the center at (0,2,0). Small cavity is a ball of radius 0.5 with the center at (0,−1,0). Connector is a cylinder of radius 0.2 with centers of its bases in the centers of the balls. A small ball inside larger cavity is centered at (0,2,0.5), and has radius 0.2.

## Mesh

Mesh grading of the rectangular box is 20×45×20 was obtained with use of blockMesh utility. Mesh of the dumbbell shaped region was obtained by SnappyHexMesh utility.

## Initial condition

The function at zero time is Gaussian distribution centered in a larger cavity
 Ψ(x,y,z) = e[1/4](−x2−(y+1)2−z2).

## Boundary conditions

Neumann (zero gradient normal to the boundary).

## Calculations

The time step is ∆t = 10−4.

## Results

Wave function stays in a smaller cavity, without considerable tunneling. The dumbbell is sliced to show the interior.