A simple a posteriori estimate on general polytopal meshes with applications to complex porous media flows

Abstract : This paper develops an posteriori error estimate for lowest-order locally conservative methods on meshes consisting of very general polytopal elements. We focus on the ease of implementation of the methodology based on H1-conforming potential reconstruction and H(div, Ω)-conforming flux reconstruction. In particular, the evaluation of our estimates merely consists in some local matrix-vector multiplications, where, on each mesh element, the matrices are either directly inherited from the given numerical method, or trivially constructed from the element geometry, while the vectors are the degrees of freedom on the given element. We then apply this methodology to unsteady nonlinear coupled degenerate problems describing complex multiphase flows in porous media. Here, on each step of the time-marching scheme, linearization procedure, and linear algebraic solver, we distinguish the corresponding error components. This leads to an easy-to-implement and fast-to-run adaptive algorithm with simultaneously guaranteed overall precision and optimal efficiency ensured through the use of adaptive stopping criteria together with adaptive space and time mesh refinements. Numerous numerical experiments on practical problems in two and three space dimensions illustrate the performance of our methodology.
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Martin Vohralík, Soleiman Yousef. A simple a posteriori estimate on general polytopal meshes with applications to complex porous media flows. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2018, 331, pp.728-760. ⟨http://www.sciencedirect.com/science/article/pii/S0045782517307442⟩. ⟨10.1016/j.cma.2017.11.027⟩. ⟨hal-01532195v3⟩

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