A Posteriori Error Estimation and Adaptive Meshing

A finite element solution $u_h$ satisfies the discrete equations exactly. What remains is the residual of the continuous problem, the functional

$$r(v)=\ell (v)-a(u_h,v)={\int }_{\partial \Omega }gv\mathrm{d}s-{\int }_{\Omega }\sigma \nabla u_h\cdot \nabla v\mathrm{d}x,$$

which vanishes for $v\in V_h$ (Galerkin orthogonality) but not for general $v\in H^1$. Note that the algebraic residual $\mathbf{g}-L_{\sigma }{\mathbf{u}}_h$ is zero, up to solver tolerance, and says nothing about discretisation error.

Residual-based estimator. Integrating by parts cell-wise gives local indicators

$${\eta }_K^2=h_K^2\Vert \nabla \cdot (\sigma \nabla u_h){\Vert }_{L^2(K)}^2+\sum _{F\subset \partial K}{h}_F\Vert [[\sigma {\partial }_{\nu }{u}_h]]{\Vert }_{L^2(F)}^2,$$

where the second term contains the jumps of the normal flux across interior faces. On boundary faces the jump is replaced by $g-\sigma {\partial }_{\nu }{u}_h$. Then $\Vert u-u_h{\Vert }_a\lesssim ({\sum }_K{\eta }_K^2{)}^{1/2}$ (reliability), and ${\eta }_K$ also bounds the local error from below (efficiency). The boundary residuals of the electrode models are listed in Residual Estimator for the Conductivity Equation.

Adaptive loop. SOLVE → ESTIMATE → MARK (for example Dörfler Marking: the smallest set of cells carrying a fixed fraction of the total error) → REFINE ($h$-refinement by splitting, with Hanging Nodes on quadrilaterals or Newest Vertex Bisection on triangles, or $p$-refinement by raising the polynomial degree). Recovery-based estimators such as the Zienkiewicz-Zhu Estimator are a cheap alternative to residual estimators.

In EIT. Errors concentrate near electrodes, where currents are singular at the electrode edges, and at conductivity jumps. Goal-oriented (dual-weighted residual) estimators, which weight residuals with the adjoint solution, target exactly the error in the measured boundary voltages. See Adaptive Meshing in EIT.

In ModularEIT.jl: AdaptiveMesh, residual_indicator.

References

1. M. Ainsworth, J. T. Oden (2000). A Posteriori Error Estimation in Finite Element Analysis. Wiley. doi:10.1002/9781118032824
2. R. Becker, R. Rannacher (2001). An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica 10, 1–102. doi:10.1017/S0962492901000010
3. W. Dörfler (1996). A Convergent Adaptive Algorithm for Poisson’s Equation. SIAM J. Numer. Anal. 33(3), 1106–1124. doi:10.1137/0733054