Solving DC Equations

(338)\[\boldsymbol{\nabla} \cdot \sigma\boldsymbol{\nabla}\phi = \boldsymbol{\nabla}\cdot\mathbf{j}_{source}.\]



For an arbitrary conductivity model, equation (338) cannot be solved exactly. In order to simulate a geophysical survey over an earth with a complicated conductivity distribution we need to solve an approximate discrete form of this equation.

The equation can be discretized directly using, for example, standard finite difference, finite element, or finite volume methods. However if we use a mimetic discretization of the full Maxwell equations, we can derive a discretization of the DC equation from the discrete Maxwell equations. A brief overview of this approach can be found in Solving Maxwell’s Equations. The following notation for the discrete system in this section comes from that page.

The discrete potential field condition is \(\tilde{\mathbf{e}} = \mathbf{G}\tilde{\phi}\). Substituting that into the discrete time-domain quasi-static Ampere equation gives

\[\mathbf{C}^T \mathbf{M}_{\mu^{-1}}^f \tilde{\mathbf{b}} - \mathbf{M}_{\sigma}^e\mathbf{G}\tilde{\phi} = \tilde{\mathbf{s}},\]

where the tilde symbol denotes a grid function. Using the fact that the discrete divergence operator is equal to \(-\mathbf{G}^T\), we take the discrete divergence of Ampere’s law to get

(339)\[-\mathbf{G}^T\mathbf{C}^T \mathbf{M}_{\mu^{-1}}^f \tilde{\mathbf{b}} + \mathbf{G}^T\mathbf{M}_{\sigma}^e\mathbf{G}\tilde{\phi} = - \mathbf{G}^T\tilde{\mathbf{s}}.\]

Since we used a mimetic discretization method, \(\mathbf{G}^T\mathbf{C}^T\) is identically zero, which corresponds the vector calculus identity \(\boldsymbol{\nabla\cdot}\left(\boldsymbol{\nabla\times}\mathbf{b}\right) = 0\). Hence the first term of equation (339) vanishes, which yields the discrete DC potential equation

(340)\[\mathbf{G}^T\mathbf{M}_{\sigma}^e\mathbf{G} \tilde{\phi} = -\mathbf{G}^T\tilde{\mathbf{s}}.\]