Some errors estimates for the box method
SIAM Journal on Numerical Analysis
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
SIAM Journal on Scientific and Statistical Computing
On the implementation of mixed methods as nonconforming methods for second-order elliptic problems
Mathematics of Computation
Finite Element Approximation of the Diffusion Operator on Tetrahedra
SIAM Journal on Scientific Computing
Explicit error bounds in a conforming finite element method
Mathematics of Computation
SIAM Journal on Scientific Computing
A Local A Posteriori Error Estimator Based on Equilibrated Fluxes
SIAM Journal on Numerical Analysis
Robust A Posteriori Error Estimation for Nonconforming Finite Element Approximation
SIAM Journal on Numerical Analysis
Robust A Posteriori Error Estimates for Stationary Convection-Diffusion Equations
SIAM Journal on Numerical Analysis
A posteriori estimation of the linearization error for strongly monotone nonlinear operators
Journal of Computational and Applied Mathematics
SIAM Journal on Numerical Analysis
Recovery-Based Error Estimator for Interface Problems: Conforming Linear Elements
SIAM Journal on Numerical Analysis
Journal of Computational and Applied Mathematics
Journal of Scientific Computing
Journal of Computational and Applied Mathematics
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We study in this paper a posteriori error estimates for H 1-conforming numerical approximations of diffusion problems with a diffusion coefficient piecewise constant on the mesh cells but arbitrarily discontinuous across the interfaces between the cells. Our estimates give a global upper bound on 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. They are guaranteed, meaning that they feature no undetermined constants. (Local) lower bounds for the error are also derived. Herein, only generic constants independent of the diffusion coefficient appear, whence our estimates are fully robust with respect to the jumps in the diffusion coefficient. In particular, no condition on the diffusion coefficient like its monotonous increasing along paths around mesh vertices is imposed, whence the present results also include the cases with singular solutions. For the energy error setting, the key requirement turns out to be that the diffusion coefficient is piecewise constant on dual cells associated with the vertices of an original simplicial mesh and that harmonic averaging is used in the scheme. This is the usual case, e.g., for the cell-centered finite volume method, included in our analysis as well as the vertex-centered finite volume, finite difference, and continuous piecewise affine finite element ones. For the dual norm setting, no such a requirement is necessary. Our estimates are based on H(div)-conforming flux reconstruction obtained thanks to the local conservativity of all the studied methods on the dual grids, which we recall in the paper; mutual relations between the different methods are also recalled. Numerical experiments are presented in confirmation of the guaranteed upper bound, full robustness, and excellent efficiency of the derived estimators.