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A posteriori error estimates including algebraic error and stopping criteria for iterative solvers. (English) Zbl 1215.65168

The authors consider the finite volume discretization of the second-order elliptic pure diffusion model problem \(-\nabla \cdot \left( {\mathbf S}\nabla p\right) =0\) in \(\Omega ,\) \(p=g\) on \(\partial \Omega .\) The first goal of the paper is to derive a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system. The derived upper bound consists of three estimators: an estimator reflecting the discretization error, an estimator corresponding to the interpolation error in the approximation of the source term and an abstract algebraic error estimator corresponding to the inexact solution of the discrete linear algebraic problem.
The authors prove that the algebraic error can be bounded by constructing an equilibrated Raviart-Thomas-Nédélec discrete vector field whose divergence is given by a proper weighting of the residual vector. On the basis of a comparison of the discretization and algebraic error estimates, they construct efficient stopping criteria for iterative algebraic solvers. Under the assumption of a convenient balance between the two estimates, they prove the local efficiency of the estimates. Thus, they correctly predict the overall error size and distribution and are suitable for adaptive mesh refinement which takes into account the inaccuracy of the algebraic computations. Numerical experiments illustrate the proposed estimates and the construction of efficient stopping criteria for algebraic iterative solvers.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65N08 Finite volume methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs

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