Residual Estimator for the Conductivity Equation

The residual-based estimator of A Posteriori Error Estimation and Adaptive Meshing, written out for the discretised Conductivity Equation with the boundary conditions of the Electrode Models. For a discrete state $u_h$ and a current pattern with prescribed boundary current density $g$:

$${\eta }_K^2=h_K^2\Vert \nabla \cdot (\sigma \nabla u_h){\Vert }_{L^2(K)}^2+\frac{1}{2}\sum _{F\subset \partial K\setminus \partial \Omega }{h}_F\Vert [[\sigma {\partial }_{\nu }{u}_h]]{\Vert }_{L^2(F)}^2+\sum _{F\subset \partial K\cap \partial \Omega }{h}_F\Vert g-\sigma {\partial }_{\nu }{u}_h{\Vert }_{L^2(F)}^2.$$

With several current patterns, the indicators are summed, because one mesh serves all of them.

Element residual. $\nabla \cdot (\sigma \nabla u_h)=\nabla \sigma \cdot \nabla u_h+\sigma \Delta u_h$ inside a cell. It vanishes for linear elements with a piecewise constant conductivity, and for continuous $\sigma$ or higher-order elements it does not.

Jumps of the current density. The exact normal current density $\sigma {\partial }_{\nu }u$ is continuous across every facet, also where $\sigma$ jumps. It is therefore the current density, not the gradient, whose jump is measured, and a correct kink of $u$ at a conductivity interface is not flagged. On meshes with Hanging Nodes, each fine piece of a coarse facet is a facet of its own. There the coarse cell’s current density is evaluated at the fine facet’s quadrature points.

Boundary residuals of the electrode models. The prescribed density $g$ comes from the electrode model:

model	$g$ on an electrode $e_{\ell }$	$g$ in the gaps
continuum	the applied current density	—
gap	$I_{\ell }/|e_{\ell }|$	$0$
CEM	$(U_{\ell }-u_h)/z_{\ell }$	$0$
point	point source (not estimated)	$0$

On Dirichlet boundaries, as in voltage-driven problems or the Shunt Model electrodes, there is no boundary residual. The CEM residual includes the Robin condition, so it detects the steep boundary layers under electrodes with a small contact impedance.

Reliability and efficiency. $\Vert {\sigma }^{1/2}\nabla (u-u_h){\Vert }_{L^2}\le C({\sum }_K{\eta }_K^2{)}^{1/2}$, and each ${\eta }_K$ bounds the error on a patch around $K$ from below, up to data oscillation. With strongly discontinuous $\sigma$, the constants depend on the contrast unless the terms are weighted with the local conductivities (Bernardi and Verfürth; Petzoldt).

Goal-oriented use. Products of the indicators of the state and of the measurement duals give a residual-based goal-oriented indicator for the electrode voltages. This is the residual counterpart of the product of recovery estimates.

In ModularEIT.jl: residual_indicator.

References

1. I. Babuška, W. C. Rheinboldt (1978). Error Estimates for Adaptive Finite Element Computations. SIAM J. Numer. Anal. 15(4), 736–754. doi:10.1137/0715049
2. C. Bernardi, R. Verfürth (2000). Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85(4), 579–608. doi:10.1007/PL00005393
3. M. Petzoldt (2002). A Posteriori Error Estimators for Elliptic Equations with Discontinuous Coefficients. Adv. Comput. Math. 16(1), 47–75. doi:10.1023/A:1014221125034
4. M. Ainsworth, J. T. Oden (2000). A Posteriori Error Estimation in Finite Element Analysis. Wiley. doi:10.1002/9781118032824