Standalone libCEED

The following two 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 examples/.

Ex1-Volume

This example is located in the subdirectory 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:

(2)\[ I = \int_{\Omega} 1 \, dV . \]

Using the same notation as in Theoretical Framework, we write here the vector \(u(x)\equiv 1\) in the Galerkin approximation, and find the volume of \(\Omega\) as

(3)\[ \sum_e \int_{\Omega_e} v(x) 1 \, dV \]

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

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

(4)\[ I = \int_{\partial \Omega} 1 \, dS , \]

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 (3), we can evaluate the surface integral by applying the volumetric Laplacian as follows

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