NDSolve`FEM`
NDSolve`FEM`

# ElementMeshInterpolation

ElementMeshInterpolation[{emesh},{f1,f2,}]

constructs an InterpolatingFunction object of the function values fj, corresponding to coordinate j of an ElementMesh object.

ElementMeshInterpolation[{{t1,t2,},emesh},
{{{f11,f12,}},{{f21,f22,}},}]

constructs an interpolation of the function values fij, corresponding to discrete ti and coordinate j of an ElementMesh object.

# Details and Options

• ElementMeshInterpolation returns an InterpolatingFunction object, which can be used like any other pure function.
• The function values f can be real or complex numbers.
• The ElementMesh object may contain curved elements.
• ElementMeshInterpolation works by fitting polynomial curves between data points belonging to the same element.
• The degree of the polynomial curves is specified by the option InterpolationOrder.
• The default InterpolationOrder is the order of the ElementMesh.
• You can do linear interpolation by using the setting .
• ElementMeshInterpolation[data] generates an InterpolatingFunction object that returns values with MachinePrecision.
• ElementMeshInterpolation has the same options as Interpolation, with the following additions:
•  "ExtrapolationHandler" Automatic specify how extrapolation is handled
• Setting the option from NDSolve and related functions is explained in NDSolve Finite Element Options.

# Examples

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## Basic Examples(2)

Set up an ElementMesh:

Set up function values at the mesh coordinates:

Create an InterpolatingFunction:

Apply the function to find interpolated values:

Plot the interpolating function:

Generate function values for a time-dependent interpolation:

Construct a time-dependent interpolating function:

## Options(5)

### "ExtrapolationHandler"(5)

Construct an InterpolatingFunction:

Query the InterpolatingFunction outside of its domain:

Construct an InterpolatingFunction with an extrapolation handler that returns Indeterminate for queries outside the domain:

Query the InterpolatingFunction outside of its domain:

Construct an InterpolatingFunction with an extrapolation handler that returns 0 outside its domain:

Query the InterpolatingFunction outside of its domain:

Construct an InterpolatingFunction with an extrapolation handler that returns Indeterminate outside its domain and does not give a warning message:

Query the InterpolatingFunction outside of its domain:

The default for NDSolve and the finite element method is to return InterpolatingFunction objects that do not extrapolate outside of the given domain:

Allow InterpolatingFunction objects generated by NDSolve to extrapolate when evaluated outside of the simulation domain and not warn about it:

While it is not generally possible to construct periodic interpolating functions for arbitrary meshes, one can mimic periodic interpolating functions based on rectangular regions by specifying an "ExtrapolatiopnHandler".

Generate data and a mesh:

Generate a temporary interpolating function from this mesh and data:

Use the extrapolation handler to map the coordinates outside of the meshed domain back into the domain and evaluate over the temporary interpolating function:

To verify that the interpolation is now periodic, visualize the function f both inside and outside the original domain:

## Properties & Relations(1)

For time-independent interpolation, ListInterpolation can also be used:

Wolfram Research (2020), ElementMeshInterpolation, Wolfram Language function, https://reference.wolfram.com/language/FEMDocumentation/ref/ElementMeshInterpolation.html (updated 2021).

#### Text

Wolfram Research (2020), ElementMeshInterpolation, Wolfram Language function, https://reference.wolfram.com/language/FEMDocumentation/ref/ElementMeshInterpolation.html (updated 2021).

#### CMS

Wolfram Language. 2020. "ElementMeshInterpolation." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/FEMDocumentation/ref/ElementMeshInterpolation.html.

#### APA

Wolfram Language. (2020). ElementMeshInterpolation. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/FEMDocumentation/ref/ElementMeshInterpolation.html

#### BibTeX

@misc{reference.wolfram_2024_elementmeshinterpolation, author="Wolfram Research", title="{ElementMeshInterpolation}", year="2021", howpublished="\url{https://reference.wolfram.com/language/FEMDocumentation/ref/ElementMeshInterpolation.html}", note=[Accessed: 24-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_elementmeshinterpolation, organization={Wolfram Research}, title={ElementMeshInterpolation}, year={2021}, url={https://reference.wolfram.com/language/FEMDocumentation/ref/ElementMeshInterpolation.html}, note=[Accessed: 24-June-2024 ]}