simpeg.electromagnetics.time_domain.Simulation3DElectricField.getAdc

Simulation3DElectricField.getAdc()

The system matrix for the DC resistivity problem.

The solution to the DC resistivity problem is necessary at the initial time for galvanic sources whose currents are non-zero at the initial time. The discrete solution to the 3D DC resistivity problem is expressed as:

\[\mathbf{A_{dc}}\,\boldsymbol{\phi_0} = \mathbf{q_{dc}}\]

where \(\mathbf{A_{dc}}\) is the DC resistivity system matrix, \(\boldsymbol{\phi_0}\) is the discrete solution for the electric potentials at the initial time, and \(\mathbf{q_{dc}}\) is the galvanic source term. This method returns the system matrix for the nodal formulation, i.e.:

\[\mathbf{A_{dc}} = \mathbf{G^T \, M_{e\sigma} \, G}\]

where \(\mathbf{G}\) is the nodal gradient operator with imposed boundary conditions, and \(\mathbf{M_{e\sigma}}\) is the inner product matrix for conductivities projected to edges.

The electric fields at the initial time \(\mathbf{e_0}\) are obtained by applying the nodal gradient operator. I.e.:

\[\mathbf{e_0} = \mathbf{G} \, \boldsymbol{\phi_0}\]

See the Notes section of the doc strings for resistivity.Simulation3DNodal for a full description of the nodal DC resistivity formulation.

Returns:

(n_nodes, n_nodes) sp.sparse.csr_matrix

The system matrix for the DC resistivity problem.