Common notation

For most of our examples, the spatial discretization uses high-order finite elements/spectral elements, namely, the high-order Lagrange polynomials defined over \(P\) non-uniformly spaced nodes, the Gauss-Legendre-Lobatto (GLL) points, and quadrature points \(\{q_i\}_{i=1}^Q\), with corresponding weights \(\{w_i\}_{i=1}^Q\) (typically the ones given by Gauss or Gauss-Lobatto quadratures, that are built in the library).

We discretize the domain, \(\Omega \subset \mathbb{R}^d\) (with \(d=1,2,3\), typically) by letting \(\Omega = \bigcup_{e=1}^{N_e}\Omega_e\), with \(N_e\) disjoint elements. For most examples we use unstructured meshes for which the elements are hexahedra (although this is not a requirement in libCEED).

The physical coordinates are denoted by \(\bm{x}=(x,y,z) \equiv (x_0,x_1,x_2) \in\Omega_e\), while the reference coordinates are represented as \(\bm{X}=(X,Y,Z) \equiv (X_0,X_1,X_2) \in \textrm{I}=[-1,1]^3\) (for \(d=3\)).