Abstract
In this article we consider the a posteriori error estimation and adaptive mesh refinement for the numerical approximation of the travel time functional arising in porous media flows. The key application of this work is in the safety assessment of radioactive waste facilities; in this setting, the travel time functional measures the time taken for a nonsorbing radioactive solute, transported by groundwater, to travel from a potential site deep underground to the biosphere. To ensure the computability of the travel time functional, we employ a mixed formulation of Darcy's law and conservation of mass, together with Raviart--Thomas $H(\mathrm{div} , \Omega)$-conforming finite elements. The proposed a posteriori error bound is derived based on a variant of the standard dual-weighted-residual approximation, which takes into account the lack of smoothness of the underlying functional of interest. The proposed adaptive refinement strategy is tested on both a simple academic test case and a problem based on the ...
Highlights
In recent decades the use of numerical simulations in hydrogeological studies has become commonplace across a range of applications
Amongst these, modelling the post-closure safety performance of deep geological storage of radioactive waste is of particular interest for a posteriori error estimation
Interpreting the adjoint solution as a generalized Green’s function for the travel time functional and the system of partial differential equations (PDEs) given in equation (2.1), we observe that the computed numerical approximation corresponds to a δfunction type source, or sink, along Ppuh,0q
Summary
In recent decades the use of numerical simulations in hydrogeological studies has become commonplace across a range of applications. Given the difficulty in demonstrating Frechet differentiability of the travel time functional in some neighbourhood of the solution ru, Hs, we adopt a different approach to that presented in section 4.2 for Frechet differentiable functionals With this in mind, we exploit a numerical approximation of the Gateaux derivative in the definition of the corresponding adjoint problem. Interpreting the adjoint solution as a generalized Green’s function for the travel time functional and the system of PDEs given in equation (2.1), we observe that the computed numerical approximation corresponds to a δfunction type source, or sink, along Ppuh,0q This is, in a sense, qualitatively similar in character to the adjoint solutions computed for first–order hyperbolic conservation laws, when the functional of interest is a point evaluation of the primal solution, cf [30, 32], for example.
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