Mimetic Discretization of the Integration by Parts Formula

The divergence theorem states that

\[ \int_{\Omega} (\nabla \cdot \mathbf{u}) \, q \, dV = \int_{\partial \Omega} q \, (\mathbf{u} \cdot \mathbf{n}) \, ds - \int_{\Omega} \mathbf{u} \cdot \nabla q \, dV \qquad \qquad (1) \]

where the boundary integral represents the flux through \(\partial \Omega\).

In one dimension, this reduces to the familiar integration by parts (IBP) formula:

\[ \int_a^b u'(x)\, q(x)\, dx = \big[ u(x)\, q(x) \big]_a^b - \int_a^b u(x)\, q'(x)\, dx \qquad \qquad (2) \]

where the boundary term is \(\big[ u(x)\, q(x) \big]_a^b = u(b)q(b) - u(a)q(a)\).

If the boundary term vanishes (e.g., homogeneous Dirichlet or periodic boundary conditions), we obtain the following

\[ \int_a^b u'(x)\, q(x)\, dx = - \int_a^b u(x)\, q'(x)\, dx \qquad \qquad (3) \]

and let

\[ u_h \in \mathcal{F}_h, \qquad q_h \in \mathcal{C}_h \]

be discrete fields, where \(\mathcal{F}_h\) denotes the discrete space associated with face-centered (vector) quantities, and \(\mathcal{C}_h\) the space associated with cell-centered (scalar) quantities.

Define the one-dimensional mimetic operators divergence and gradient:

\[ D: \mathcal{F}_h \to \mathcal{C}_h, \qquad G: \mathcal{C}_h \to \mathcal{F}_h \]

The weighted inner products on \(\mathcal{F}_h\) and \(\mathcal{C}_h\) are induced by diagonal, positive-definite matrices \(P\) and \(Q\), respectively.

Then, the discrete analog of (3) is given by

\[ \langle D u_h, q_h \rangle_{Q} = - \langle u_h, G q_h \rangle_{P}, \qquad \forall\, u_h, q_h \]

or, in matrix form,

\[ (D u_h)^{T} Q\, q_h = -\, u_h^{T} P\, (G q_h) \]

The example integration1D.m illustrates how the weight matrix \(Q\) can be used to approximate the integral of a Ricker wavelet.