# Standalone libCEED

The following three examples have no dependencies, and are designed to be self-contained.
For additional examples that use external discretization libraries (MFEM, PETSc, Nek5000 etc.) see the subdirectories in {file}`examples/`.

(ex1-volume)=

## Ex1-Volume

This example is located in the subdirectory {file}`examples/ceed`.
It illustrates a simple usage of libCEED to compute the volume of a given body using a matrix-free application of the mass operator.
Arbitrary mesh and solution orders in 1D, 2D, and 3D are supported from the same code.

This example shows how to compute line/surface/volume integrals of a 1D, 2D, or 3D domain $\Omega$ respectively, by applying the mass operator to a vector of $1$s.
It computes:

$$
I = \int_{\Omega} 1 \, dV .
$$ (eq-ex1-volume)

Using the same notation as in {ref}`theoretical-framework`, we write here the vector $u(x)\equiv 1$ in the Galerkin approximation, and find the volume of $\Omega$ as

$$
\sum_e \int_{\Omega_e} v(x) 1 \, dV
$$ (volume-sum)

with $v(x) \in \mathcal{V}_p = \{ v \in H^{1}(\Omega_e) \,|\, v \in P_p(\bm{I}), e=1,\ldots,N_e \}$, the test functions.

(ex2-surface)=

## Ex2-Surface

This example is located in the subdirectory {file}`examples/ceed`.
It computes the surface area of a given body using matrix-free application of a diffusion operator.
Similar to {ref}`Ex1-Volume`, arbitrary mesh and solution orders in 1D, 2D, and 3D are supported from the same code.
It computes:

$$
I = \int_{\partial \Omega} 1 \, dS ,
$$ (eq-ex2-surface)

by applying the divergence theorem.
In particular, we select $u(\bm x) = x_0 + x_1 + x_2$, for which $\nabla u = [1, 1, 1]^T$, and thus $\nabla u \cdot \hat{\bm n} = 1$.

Given Laplace's equation,

$$
\nabla \cdot \nabla u = 0, \textrm{ for  } \bm{x} \in \Omega ,
$$

let us multiply by a test function $v$ and integrate by parts to obtain

$$
\int_\Omega \nabla v \cdot \nabla u \, dV - \int_{\partial \Omega} v \nabla u \cdot \hat{\bm n}\, dS = 0 .
$$

Since we have chosen $u$ such that $\nabla u \cdot \hat{\bm n} = 1$, the boundary integrand is $v 1 \equiv v$. Hence, similar to {eq}`volume-sum`, we can evaluate the surface integral by applying the volumetric Laplacian as follows

$$
\int_\Omega \nabla v \cdot \nabla u \, dV \approx \sum_e \int_{\partial \Omega_e} v(x) 1 \, dS .
$$

(ex3-volume)=

## Ex3-Volume

This example is located in the subdirectory {file}`examples/ceed`.
It illustrates a more complex usage of libCEED to compute the volume of a given body using a matrix-free application of the screened Poisson operator.
Arbitrary mesh and solution orders in 1D, 2D, and 3D are supported from the same code.

This example shows how to compute line/surface/volume integrals of a 1D, 2D, or 3D domain $\Omega$ respectively, by applying the screened Poisson operator to a vector of $1$s.
It computes:

$$
I = \int_{\Omega} \left( 1 + \nabla^2 1 \right) \, dV .
$$ (eq-ex3-volume)

Using the same notation as in {ref}`theoretical-framework`, we write here the vector $u(x)\equiv 1$ in the Galerkin approximation, and find the volume of $\Omega$ as

$$
\sum_e \int_{\Omega_e}\left( v(x) 1 + \nabla v(x) \cdot 0 \right) \, dV
$$ (volume-sum)

with $v(x) \in \mathcal{V}_p = \{ v \in H^{1}(\Omega_e) \,|\, v \in P_p(\bm{I}), e=1,\ldots,N_e \}$, the test functions.

The addition of the Poisson term is not needed to compute the volume of the region, as shown in example 1.
Rather, this example illustrates the ability to add multiple evaluation modes for the same input or output vector in a libCEED operator.