Find 5 eigenvalues and eigenvectors of a Laplacian with periodic boundary conditions:

Compare the eigenvalues with the expected analytical eigenvalues:

Visualize the eigenfunctions:

Inspect to see that the bounds are periodic:

Solve a Poisson equation with periodic boundary conditions on curved boundaries:

Visualize the solution:

Visualize the periodic solution:

Compute eigenvalues of a Laplacian in polar coordinates over a Disk:

Analytically compute the eigenvalues of a Laplacian in Cartesian coordinates over a Disk:

Compare the eigenvalues computed:

Numerically compute the eigenvalues of a Laplacian in Cartesian coordinates over a Disk:

Compare the numerically computed eigenvalues computed in Cartesian coordinates with the analytical solution:

Compare the numerically computed eigenvalues:

Exclusively specifying periodic boundary condition for stationary PDEs may result in a non-unique solution. In some cases, the system may not be solvable at all:

Specify a DirichletCondition on the source boundary:

The source for the mapping f of the PeriodicBoundaryCondition need not necessarily be on the boundary of a domain:

Visualize the result:

Note that :

Compare to the solution where the source is on the boundary:

Visualize the result:

DirichletCondition cannot be present on the target boundary:

Moving the DirichletCondition to the source boundary works:

Periodic boundary conditions propagate the solution of a PDE from the source to the target boundary. The choice of which boundary is considered source and which is target has an influence on the solution of the PDE.

Consider a rectangular domain and an asymmetric Poisson equation:

The lower and upper parts are subject to a DirichletCondition:

First, the left-hand edge is the target boundary to the right-hand edge (the source):

Solve and visualize the equation:

Inspect the values at the periodic boundaries:

In the second example, the right-hand edge is a target boundary to the left-hand edge (the source):

Solve and visualize the equation:

Inspect the values at the periodic boundaries:

Periodic boundary conditions propagate the solution of a PDE from a source region to the target region. The direction of the mapping has an influence on the solution of the PDE.

Consider a rectangular domain and an asymmetric Poisson equation:

The lower and upper parts are subject to a DirichletCondition:

Map the lower right to the higher left-hand side and the higher right to the lower left-hand side:

The left-hand side is a target boundary to the right-hand side:

Solve and visualize the equation:

Compare to mapping in the same direction but with flipped edges; the lower right to the lower left-hand side and the higher right to the higher left-hand side:

The left-hand side is a target boundary to the right-hand side:

Solve and visualize the equation:

Periodic boundary conditions relate the solution of a PDE from the source to the target boundary. Boundary conditions present, also implicit ones, at the source will affect the solution at the target.

To exemplify the behavior, consider a time-dependent equation discretized with the finite element method. An initial condition u, implicit Neumann zero boundary conditions on both sides and no PeriodicBoundaryCondition are specified:

Visualize the solution at various times:

Note that at both spatial boundaries the implicit Neumann 0 boundary conditions are satisfied.

When a PeriodicBoundaryCondition is used on a source boundary that has an implicit Neumann 0 boundary condition, then that condition will be mapped to the target boundary.

Following is the solution of the same equation and initial condition as previously and an additional periodic boundary condition that has its source on the left and its target on the right:

Visualize the solution at various times:

Note how the solution value at the implicit Neumann 0 boundary condition on the left is mapped to the right.

This is the expected behavior for the finite element method. The tensor product grid method behaves differently, as that method does not have implicit boundary conditions:

Visualize the tensor product grid solution at various times:

A similar behavior can be achieved with the finite element method by specifying a DirichletCondition on the left and a PeriodicBoundaryCondition:

Visualize the difference between the finite element and tensor product grid solutions at various times:

Alternatively, a DirichletCondition could be specified at each side.

Periodic boundary conditions relate the solution of a PDE from the source to the target boundary. Boundary conditions present, including implicit ones, at the source will affect the solution at the target. Multiple periodic boundary conditions can be used to enforce continuous derivatives.

Set up a region and a helper function that takes periodic boundary conditions as arguments:

Set up plotting options and a helper function to create periodic plots:

Solve the PDE with a single periodic boundary condition on the left-hand side:

Visualize a periodic contour plot of the solution:

Visualize a periodic stream density plot of the gradient of the solution:

Note that the gradient is not continuous. This because of the implicit Neumann value that is present at the source location.

Similarly, visualize a periodic density plot of the norm of the gradient of the solution:

Specifying a second periodic boundary condition enforces a continuous derivative. Solve the PDE with periodic boundary conditions on both the left- and right-hand sides:

Visualize a periodic contour plot of the solution. As expected, the solution changes, since the implicit Neumann value is now eliminated:

Visualize a periodic stream density plot of the gradient of the solution. Note that now the gradient is continuous:

Similarly, visualize a periodic density plot of the norm of the gradient of the solution and find that the norm of the gradient is also continuous:

This approach works on unstructured meshes.

Periodic boundary conditions, like Neumann values, operate on edges in 2D and surfaces in 3D. This means that sufficient nodes need to be present in a finite element mesh to make periodic boundary conditions work.

Generate a finite element mesh and visualize the mesh nodes:

Solve a PDE with a periodic boundary condition that maps a single point:

To fix this, specify a predicate for the periodic boundary condition that covers an edge: