## Dimensions of Solutions, Gradients, and Fluxes

`solvepde`

returns a `StationaryResults`

or `TimeDependentResults`

object whose properties contain the solution and its
gradient at the mesh nodes. You can interpolate the solution and its gradient to other
points in the geometry by using `interpolateSolution`

and `evaluateGradient`

. You also can compute flux of the solution at the mesh
nodes and at arbitrary points by using `evaluateCGradient`

.

**Note**

`solvepde`

does not compute components of
flux of a PDE solution. To compute flux of the solution at the mesh
nodes, use `evaluateCGradient`

.

`solvepdeeig`

returns an `EigenResults`

object whose properties contain the solution eigenvectors
calculated at the mesh nodes. You can interpolate the solution to other points by using
`interpolateSolution`

.

The dimensions of the solution, its gradient, and flux of the solution depend on:

The number of geometric evaluation points.

For results returned by

`solvepde`

or`solvepdeeig`

, this is the number of mesh nodes.For results returned by

`interpolateSolution`

,`evaluateGradient`

, and`evaluateCGradient`

this is the number of query points.

The number of equations.

For results returned by

`solvepde`

or`solvepdeeig`

, this is the number of equations in the system.For results returned by

`interpolateSolution`

,`evaluateGradient`

, and`evaluateCGradient`

, this is the number of query equation indices.

The number of times for a time-dependent problem or number of modes for an eigenvalue problem.

For results returned by

`solvepde`

, this is the number of solution times (specified as an input to`solvepde`

).For results returned by

`solvepdeeig`

, this is the number of eigenvalues.For results returned by

`interpolateSolution`

,`evaluateGradient`

, and`evaluateCGradient`

, this is the number of query times for time-dependent problems or query modes for eigenvalue problems.

Suppose you have a problem in which:

`Np`

is the number of nodes in the mesh.`Nt`

is the number of times for a time-dependent problem or number of modes for an eigenvalue problem.`N`

is the number of equations in the system.

Suppose you also compute the solution, its gradient, or flux of the solution at other points
("query points") in the geometry by using `interpolateSolution`

,
`evaluateGradient`

, or `evaluateCGradient`

,
respectively. Here:

`Nqp`

is the number of query points.`Nqt`

is the number of query times for a time-dependent problem or number of query modes for an eigenvalue problem.`Nq`

is the number of query equations indices.

The tables show how to index into the solution returned by `solvepde`

or `solvepdeeig`

,
where:

`iP`

contains the indices of nodes.`iT`

contains the indices of times for a time-dependent problem or mode numbers for an eigenvalue problem.`iN`

contains the indices of equations.

The tables also show the dimensions of solutions, gradients,
and flux of the solution at nodal locations (returned by `solvepde`

,`solvepdeeig`

,
and `evaluateCGradient`

) and the dimensions of
interpolated solutions and gradients (returned by `interpolateSolution`

, `evaluateGradient`

,
and `evaluateCGradient`

).

Stationary PDE problem | Access solution and components of gradient | Size of `NodalSolution` , `XGradients` , `YGradients` , `ZGradients` ,
and components of flux at nodal points | Size of solution, components of gradient, and components of flux at query points |
---|---|---|---|

Scalar |
| `Np` -by-1 | `Nqp` -by-1 |

System, `N` > `1` |
| `Np` -by-`N` | `Nqp` -by-`N` |

Time-dependent PDE problem | Access solution and components of gradient | Size
of `NodalSolution` , `XGradients` , `YGradients` , `ZGradients` ,
and components of flux at nodal points | Size of solution, components of gradient, and components of flux at query points |
---|---|---|---|

Scalar |
| `Np` -by-`Nt` | `Nqp` -by-`Nqt` |

System, `N` > `1` |
| `Np` -by-`N` -by-`Nt` | `Nqp` -by-`Nq` -by-`Nqt` |

PDE eigenvalue problem | Access eigenvectors | Size of `Eigenvectors` | Size of interpolated eigenvectors |
---|---|---|---|

Scalar |
| `Np` -by-`Nt` | `Nqp` -by-`Nqt` |

System, `N` > `1` |
| `Np` -by-`N` -by-`Nt` | `Nqp` -by-`Nq` -by-`Nqt` |

## See Also

`solvepde`

| `solvepdeeig`

| `interpolateSolution`

| `evaluateGradient`

| `StationaryResults`

| `TimeDependentResults`

| `EigenResults`