A Note on a Posteriori Error Bounds for Numerical Solutions of Elliptic Equations with a Piecewise Constant Reaction Coefficient Having Large Jumps

Abstract: We have derived guaranteed, robust, and fully computable a posteriori error bounds for approximate solutions of the equation ΔΔu + k²u = f, where the coefficient k ≥ 0 is a constant in each subdomain (finite element) and chaotically varies between subdomains in a sufficiently wide range. For finite element solutions, these bounds are robust with respect to k ∈[0, ch⁻²] c = const, and possess some other good features. The coefficients in front of two typical norms on their right-hand sides are only insignificantly worse than those obtained earlier for k ≡ const. The bounds can be calculated without resorting to the equilibration procedures, and they are sharp for at least low-order methods. The derivation technique used in this paper is similar to the one used in our preceding papers (2017–2019) for obtaining a posteriori error bounds that are not improvable in the order of accuracy.