1*bcb2dfaeSJed Brown# Standalone libCEED 2*bcb2dfaeSJed Brown 3*bcb2dfaeSJed BrownThe following two examples have no dependencies, and are designed to be self-contained. 4*bcb2dfaeSJed BrownFor additional examples that use external discretization libraries (MFEM, PETSc, Nek5000 5*bcb2dfaeSJed Brownetc.) see the subdirectories in {file}`examples/`. 6*bcb2dfaeSJed Brown 7*bcb2dfaeSJed Brown(ex1-volume)= 8*bcb2dfaeSJed Brown 9*bcb2dfaeSJed Brown## Ex1-Volume 10*bcb2dfaeSJed Brown 11*bcb2dfaeSJed BrownThis example is located in the subdirectory {file}`examples/ceed`. It illustrates a 12*bcb2dfaeSJed Brownsimple usage of libCEED to compute the volume of a given body using a matrix-free 13*bcb2dfaeSJed Brownapplication of the mass operator. Arbitrary mesh and solution orders in 1D, 2D, and 3D 14*bcb2dfaeSJed Brownare supported from the same code. 15*bcb2dfaeSJed Brown 16*bcb2dfaeSJed BrownThis example shows how to compute line/surface/volume integrals of a 1D, 2D, or 3D 17*bcb2dfaeSJed Browndomain $\Omega$ respectively, by applying the mass operator to a vector of 18*bcb2dfaeSJed Brown$1$s. It computes: 19*bcb2dfaeSJed Brown 20*bcb2dfaeSJed Brown$$ 21*bcb2dfaeSJed BrownI = \int_{\Omega} 1 \, dV . 22*bcb2dfaeSJed Brown$$ (eq-ex1-volume) 23*bcb2dfaeSJed Brown 24*bcb2dfaeSJed BrownUsing the same notation as in {ref}`theoretical-framework`, we write here the vector 25*bcb2dfaeSJed Brown$u(x)\equiv 1$ in the Galerkin approximation, 26*bcb2dfaeSJed Brownand find the volume of $\Omega$ as 27*bcb2dfaeSJed Brown 28*bcb2dfaeSJed Brown$$ 29*bcb2dfaeSJed Brown\sum_e \int_{\Omega_e} v(x) 1 \, dV 30*bcb2dfaeSJed Brown$$ (volume-sum) 31*bcb2dfaeSJed Brown 32*bcb2dfaeSJed Brownwith $v(x) \in \mathcal{V}_p = \{ v \in H^{1}(\Omega_e) \,|\, v \in P_p(\bm{I}), e=1,\ldots,N_e \}$, 33*bcb2dfaeSJed Brownthe test functions. 34*bcb2dfaeSJed Brown 35*bcb2dfaeSJed Brown(ex2-surface)= 36*bcb2dfaeSJed Brown 37*bcb2dfaeSJed Brown## Ex2-Surface 38*bcb2dfaeSJed Brown 39*bcb2dfaeSJed BrownThis example is located in the subdirectory {file}`examples/ceed`. It computes the 40*bcb2dfaeSJed Brownsurface area of a given body using matrix-free application of a diffusion operator. 41*bcb2dfaeSJed BrownSimilar to {ref}`Ex1-Volume`, arbitrary mesh and solution orders in 1D, 2D, and 3D 42*bcb2dfaeSJed Brownare supported from the same code. It computes: 43*bcb2dfaeSJed Brown 44*bcb2dfaeSJed Brown$$ 45*bcb2dfaeSJed BrownI = \int_{\partial \Omega} 1 \, dS , 46*bcb2dfaeSJed Brown$$ (eq-ex2-surface) 47*bcb2dfaeSJed Brown 48*bcb2dfaeSJed Brownby applying the divergence theorem. 49*bcb2dfaeSJed BrownIn 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*bcb2dfaeSJed Brown 51*bcb2dfaeSJed BrownGiven Laplace's equation, 52*bcb2dfaeSJed Brown 53*bcb2dfaeSJed Brown$$ 54*bcb2dfaeSJed Brown\nabla \cdot \nabla u = 0, \textrm{ for } \bm{x} \in \Omega , 55*bcb2dfaeSJed Brown$$ 56*bcb2dfaeSJed Brown 57*bcb2dfaeSJed Brownlet us multiply by a test function $v$ and integrate by parts to obtain 58*bcb2dfaeSJed Brown 59*bcb2dfaeSJed Brown$$ 60*bcb2dfaeSJed Brown\int_\Omega \nabla v \cdot \nabla u \, dV - \int_{\partial \Omega} v \nabla u \cdot \hat{\bm n}\, dS = 0 . 61*bcb2dfaeSJed Brown$$ 62*bcb2dfaeSJed Brown 63*bcb2dfaeSJed BrownSince we have chosen $u$ such that $\nabla u \cdot \hat{\bm n} = 1$, the boundary integrand is $v 1 \equiv v$. Hence, similar to {math:numref}`volume-sum`, we can evaluate the surface integral by applying the volumetric Laplacian as follows 64*bcb2dfaeSJed Brown 65*bcb2dfaeSJed Brown$$ 66*bcb2dfaeSJed Brown\int_\Omega \nabla v \cdot \nabla u \, dV \approx \sum_e \int_{\partial \Omega_e} v(x) 1 \, dS . 67*bcb2dfaeSJed Brown$$ 68