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1 Department of Numerical Mathematics 2 CERMICS - Centre d-Enseignement et de Recherche en Mathématiques et Calcul Scientifique 3 SERENA - Simulation for the Environment : Reliable and Efficient Numerical Algorithms Inria Paris-Rocquencourt

Abstract : We devise and study experimentally adaptive strategies driven by a posteriori error estimates to select automatically both the space mesh and the polynomial degree in the numerical approximation of diffusion equations in two space dimensions. The adaptation is based on equilibrated flux estimates. These estimates are presented here for inhomogeneous Dirichlet and Neumann boundary conditions, for spatially-varying polynomial degree, and for mixed rectangular-triangular grids possibly containing hanging nodes. They deliver a global error upper bound with constant one and, up to data oscillation, error lower bounds on element patches with a generic constant only dependent on the mesh regularity and with a computable bound. We numerically asses the estimates and several hp-adaptive strategies using the interior penalty discontinuous Galerkin method. Asymptotic exactness is observed for all the symmetric, nonsymmetric odd degrees, and incomplete variants on non-nested unstructured triangular grids for a smooth solution and uniform refinement. Exponential convergence rates are reported on nonmatching triangular grids for the incomplete version on several benchmarks with a singular solution and adaptive refinement.

Keywords : inhomogeneous boundary conditions Laplace equation a posteriori error estimate hp-refinement p-robustness discontinuous Galerkin





Autor: Vít Dolejší - Alexandre Ern - Martin Vohralík -

Fuente: https://hal.archives-ouvertes.fr/



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