* where the sum extends over those cells on which $\varphi_i$ is nonzero.
* The so-computed mass matrix is therefore diagonal.
*
- * Whether or not this particular choice of quadrature formula is sufficient
- * to retain the convergence rate of the discretization is a separate
- * question. For the usual $Q_k$ finite elements, the appropriate
- * quadrature formulas are of QGaussLobatto type.
+ * Whether or not this particular choice of quadrature formula is
+ * sufficient to retain the convergence rate of the discretization is
+ * a separate question. For the usual $Q_k$ finite elements
+ * (implemented by FE_Q and FE_DGQ), the appropriate quadrature
+ * formulas are of QGaussLobatto type. Mass lumping can also be done
+ * with FE_SimplexP_Bubbles, for example, if appropriate quadrature rules
+ * are chosen.
+ *
+ * For an example of where lumped mass matrices play a role, see step-69.
* </dd>
*
*
@f}
where
- $m_i \dealcoloneq \int_{\Omega} \phi_i \, \mathrm{d}\mathbf{x}$
- is the lumped @ref GlossMassMatrix "mass matrix"
+ is the @ref GlossLumpedMassMatrix "lumped"
+ @ref GlossMassMatrix "mass matrix"
- $\tau$ is the time step size
- $\mathbf{c}_{ij} \dealcoloneq \int_{\Omega} \nabla\phi_j\phi_i \,
\mathrm{d}\mathbf{x}$ (note that $\mathbf{c}_{ij}\in \mathbb{R}^d$)
//
// The class <code>OfflineData</code> contains pretty much all components
// of the discretization that do not evolve in time, in particular, the
- // DoFHandler, SparsityPattern, boundary maps, the lumped @ref GlossMassMatrix "mass matrix",
+ // DoFHandler, SparsityPattern, boundary maps, the
+ // @ref GlossLumpedMassMatrix "lumped" lumped
+ // @ref GlossMassMatrix "mass matrix",
// $\mathbf{c}_{ij}$ and $\mathbf{n}_{ij}$ matrices. Here, the term
// <i>offline</i> refers to the fact that all the class
// members of <code>OfflineData</code> have well-defined values
compute_diagonal() override;
/**
- * Compute the lumped @ref GlossMassMatrix "mass matrix". This is equal to the mass matrix times a
+ * Compute the @ref GlossLumpedMassMatrix "lumped mass matrix".
+ * This is equal to the mass matrix times a
* vector of all ones and is equivalent to approximating the mass matrix
* with a nodal quadrature rule.
*