### (Closed) What kind element family requires no integration to evaluate?

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I'm following a paper that solves a nonlinear PDE. The paper starts by describing the function space of the solution.

OK, so the solution here can be represented on a function space defined by Lagrange elements of the first degree. The nonlinear coefficients, which are functions of the solution, are described by a different function space. It states:

$q\left(x,y,z\right)=\sum_{n=1}^4q\left(x_n,y_n,z_n\right)\phi_n\left(x,y,z\right)$

So what kind of function space are the nonlinear coefficients represented on? In this problem, the integration over the elements is by far the most costly computation. Avoiding the integration of the nonlinear coefficients where possible would be really nice.

"

$u'\left(x,y,z,t\right)=\sum_{n=1}^N\phi_n\left(x,y,z\right)u\left(t\right)$

*The dependent variable*$u\left(x,y,z,t\right)$`u`(`x`,`y`,`z`,`t`)*is approximated by a function*$u'\left(x,y,z,t\right)$`u`'(`x`,`y`,`z`,`t`)*such that:*$u'\left(x,y,z,t\right)=\sum_{n=1}^N\phi_n\left(x,y,z\right)u\left(t\right)$

`u`'(`x`,`y`,`z`,`t`)=∑_{n=1}^{N}`ϕ`_{n}(`x`,`y`,`z`)`u`(`t`)*where*$\phi_n$`ϕ`_{n}*are piecewise linear basis functions,*$u\left(t\right)$`u`(`t`)*are unknown coefficients representing the solution, and $N$*`N`is the total number of nodal points."OK, so the solution here can be represented on a function space defined by Lagrange elements of the first degree. The nonlinear coefficients, which are functions of the solution, are described by a different function space. It states:

*"The nonlinear coefficients at a given point in time are assumed to vary linearly over each element. For example,*$q\left(u\right)$

`q`(

`u`)

*is expanded over each element as follows:*

$q\left(x,y,z\right)=\sum_{n=1}^4q\left(x_n,y_n,z_n\right)\phi_n\left(x,y,z\right)$

`q`(

`x`,

`y`,

`z`)=∑

_{n=1}

^{4}

`q`(

`x`

_{n},

`y`

_{n},

`z`

_{n})

`ϕ`

_{n}(

`x`,

`y`,

`z`)

*where $n$*

`n`represents the corners of the element.**The advantage of this is that no numerical integration is needed to evaluate these coefficients.**So what kind of function space are the nonlinear coefficients represented on? In this problem, the integration over the elements is by far the most costly computation. Avoiding the integration of the nonlinear coefficients where possible would be really nice.

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This is not related to FEniCS. Ask some mathematical FEM forum. In addition, you are confusing many concepts in the question.