1# Standalone libCEED 2 3The following two examples have no dependencies, and are designed to be self-contained. 4For additional examples that use external discretization libraries (MFEM, PETSc, Nek5000 5etc.) see the subdirectories in {file}`examples/`. 6 7(ex1-volume)= 8 9## Ex1-Volume 10 11This example is located in the subdirectory {file}`examples/ceed`. It illustrates a 12simple usage of libCEED to compute the volume of a given body using a matrix-free 13application of the mass operator. Arbitrary mesh and solution orders in 1D, 2D, and 3D 14are supported from the same code. 15 16This example shows how to compute line/surface/volume integrals of a 1D, 2D, or 3D 17domain $\Omega$ respectively, by applying the mass operator to a vector of 18$1$s. It computes: 19 20$$ 21I = \int_{\Omega} 1 \, dV . 22$$ (eq-ex1-volume) 23 24Using the same notation as in {ref}`theoretical-framework`, we write here the vector 25$u(x)\equiv 1$ in the Galerkin approximation, 26and find the volume of $\Omega$ as 27 28$$ 29\sum_e \int_{\Omega_e} v(x) 1 \, dV 30$$ (volume-sum) 31 32with $v(x) \in \mathcal{V}_p = \{ v \in H^{1}(\Omega_e) \,|\, v \in P_p(\bm{I}), e=1,\ldots,N_e \}$, 33the test functions. 34 35(ex2-surface)= 36 37## Ex2-Surface 38 39This example is located in the subdirectory {file}`examples/ceed`. It computes the 40surface area of a given body using matrix-free application of a diffusion operator. 41Similar to {ref}`Ex1-Volume`, arbitrary mesh and solution orders in 1D, 2D, and 3D 42are supported from the same code. It computes: 43 44$$ 45I = \int_{\partial \Omega} 1 \, dS , 46$$ (eq-ex2-surface) 47 48by applying the divergence theorem. 49In particular, we select $u(\bm x) = x_0 + x_1 + x_2$, for which $\nabla u = [1, 1, 1]^T$, and thus $\nabla u \cdot \hat{\bm n} = 1$. 50 51Given Laplace's equation, 52 53$$ 54\nabla \cdot \nabla u = 0, \textrm{ for } \bm{x} \in \Omega , 55$$ 56 57let us multiply by a test function $v$ and integrate by parts to obtain 58 59$$ 60\int_\Omega \nabla v \cdot \nabla u \, dV - \int_{\partial \Omega} v \nabla u \cdot \hat{\bm n}\, dS = 0 . 61$$ 62 63Since 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 64 65$$ 66\int_\Omega \nabla v \cdot \nabla u \, dV \approx \sum_e \int_{\partial \Omega_e} v(x) 1 \, dS . 67$$ 68