emg3d.core.amat_x(rx, ry, rz, ex, ey, ez, eta_x, eta_y, eta_z, zeta, hx, hy, hz)[source]

Residual with or without source term.

Compute the residual as given in [Muld06] in middle of the right column on page 636, but without the source term:

\[\mathbf{r} = V \left( \mathrm{i}\omega\mu_0 \tilde{\sigma} \mathbf{E} - \nabla \times \mu_\mathrm{r}^{-1} \nabla \times \mathbf{E} \right) .\]

The computation is carried out in a matrix-free manner; on said page 636 (or in the Background Theory of the manual) are the various steps laid out to discretize the different parts such as the involved curls. This can also be understood as the left-hand-side of \(A x = b\), as given in Equation 2 in [Muld06] (here without the cell volumes \(V\)),

\[\mathrm{i}\omega\mu_0 \tilde{\sigma} \mathbf{E} - \nabla \times \mu_\mathrm{r}^{-1} \nabla \times \mathbf{E} = - \mathrm{i} \omega \mu_0 \mathbf{J_\mathrm{s}} .\]

It can therefore be used as a matvec to create a LinearOperator, which can be passed to a solver.

It is assumed that the PEC boundary condition is applied to the electric field \(\mathbf{E}\) (ex, ey, and ez).

The residuals are subtracted in-place from rx, ry, and rz. That means that if rx, ry, and rz contain the source field, they will contain the total residual afterwards; if they are empty fields, they will contain the negative partial residuals afterwards.

rx, ry, rzndarray

Source field or pre-allocated zero residual field in x-, y-, and z-directions (emg3d.fields.Field).

ex, ey, ezndarray

Electric fields in x-, y-, and z-directions (emg3d.fields.Field).

eta_x, eta_y, eta_z, zetandarray

Volume-averaged model parameters (emg3d.models.VolumeModel).

hx, hy, hzndarray

Cell widths in x-, y-, and z-directions (emg3d.meshes.TensorMesh).