Robust a Posteriori Error Estimates for Conforming Discretizations of Diffusion Problems with Discontinuous Coefficients on Anisotropic Meshes

In this paper, we study a posteriori estimates for different numerical methods of diffusion problems with discontinuous coefficients on anisotropic meshes, in particular, which can be applied to vertex-centered and cell-centered finite volume, finite difference and piecewise linear finite element methods. Based on the stretching ratios of the mesh elements, we improve a posteriori estimates developed by Vohralík (J Sci Comput 46:397–438, 2011), which are reliable and efficient on isotropic meshes but fail on anisotropic ones (see the numerical results of the paper). Without the assumption that the meshes are shape-regular, the resulting mesh-dependent error estimators are shown to be reliable and efficient with respect to the error measured either as the energy norm of the difference between the exact and approximate solutions, or as a dual norm of the residual, as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution. In other words, they are equivalent to the estimates of Vohralík in the case of isotropic meshes and proved to be robust on anisotropic meshes as well. Based on \(\mathbf{H}(\mathrm {div})\)-conforming, locally conservative flux reconstruction, we suggest two different constructions of the equilibrated flux with the anisotropy of mesh, which is essential to the robustness of our estimates on anisotropic meshes. Numerical experiments in 2D confirm that our estimates are reliable and efficient on anisotropic meshes.