examples/ceed/index.md

3 The following three examples have no dependencies, and are designed to be self-contained.
6 (ex1-volume)=
8 ## Ex1-Volume
11 It illustrates a simple usage of libCEED to compute the volume of a given body using a matrix-free …
19 $$ (eq-ex1-volume)
21 Using the same notation as in {ref}`theoretical-framework`, we write here the vector $u(x)\equiv 1$…
24 \sum_e \int_{\Omega_e} v(x) 1 \, dV
25 $$ (volume-sum)
27 with $v(x) \in \mathcal{V}_p = \{ v \in H^{1}(\Omega_e) \,|\, v \in P_p(\bm{I}), e=1,\ldots,N_e \}$…
29 (ex2-surface)=
31 ## Ex2-Surface
34 It computes the surface area of a given body using matrix-free application of a diffusion operator.
35 Similar to {ref}`Ex1-Volume`, arbitrary mesh and solution orders in 1D, 2D, and 3D are supported fr…
40 $$ (eq-ex2-surface)
51 let us multiply by a test function $v$ and integrate by parts to obtain
54 \int_\Omega \nabla v \cdot \nabla u \, dV - \int_{\partial \Omega} v \nabla u \cdot \hat{\bm n}\, d…
57 … u \cdot \hat{\bm n} = 1$, the boundary integrand is $v 1 \equiv v$. Hence, similar to {eq}`volume…
60 \int_\Omega \nabla v \cdot \nabla u \, dV \approx \sum_e \int_{\partial \Omega_e} v(x) 1 \, dS .
63 (ex3-volume)=
65 ## Ex3-Volume
68 …omplex usage of libCEED to compute the volume of a given body using a matrix-free application of t…
76 $$ (eq-ex3-volume)
78 Using the same notation as in {ref}`theoretical-framework`, we write here the vector $u(x)\equiv 1$…
81 \sum_e \int_{\Omega_e}\left( v(x) 1 + \nabla v(x) \cdot 0 \right) \, dV
82 $$ (volume-sum-mass-diff)
84 with $v(x) \in \mathcal{V}_p = \{ v \in H^{1}(\Omega_e) \,|\, v \in P_p(\bm{I}), e=1,\ldots,N_e \}$…