Abstract
The paper is concerned with guaranteed a posteriori error estimates for a class of evolutionary problems related to poroelastic media governed by the quasi-static linear Biot equations. The system is decoupled by employing the fixed-stress split scheme, which leads to an iteratively solved semi-discrete system. The error bounds are derived by combining a posteriori estimates for contractive mappings with functional type error control for elliptic partial differential equations. The estimates are applicable to any approximation in the admissible functional space and are independent of the discretization method. They are fully computable, do not contain mesh-dependent constants, and provide reliable global estimates of the error measured in the energy norm. Moreover, they suggest efficient error indicators for the distribution of local errors and can be used in adaptive procedures.
Highlights
The problems defined in a poroelastic medium contribute to a wide range of application areas, including simulation of oil reservoirs, prediction of environmental changes, soil subsidence and liquefaction in earthquake engineering, well stability, sand production, waste deposition, hydraulic fracturing, CO2 sequestration, and understanding of the biological tissues in biomechanics
The derivation combines the estimates for contraction mappings and the functional a posteriori error majorants for elliptic problems
The obtained error bound is fully computable and independent on the discretization techniques used for the variational formulation of the Biot problem as soon as the reproduced approximations belong to admissible functional spaces
Summary
The problems defined in a poroelastic medium contribute to a wide range of application areas, including simulation of oil reservoirs, prediction of environmental changes, soil subsidence and liquefaction in earthquake engineering, well stability, sand production, waste deposition, hydraulic fracturing, CO2 sequestration, and understanding of the biological tissues in biomechanics. We turn to the functional error estimates (majorants) that are fully computable and provide guaranteed bounds of errors arising in numerical approximations The derivation of such estimates is based on functional arguments and variational formulation of the problem in question. Our approach is based on the contraction property of the iterative method [53], which is rather general and not restricted only to the fixed-stress scheme, and functional type estimates of each equation in the Biot system (see, e.g., [54]). It clarifies the arguments for choosing the optimal parameters in the iterative scheme and proves that it is a contraction with an explicitly computable convergence rate.
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