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- Larson, Mats G, et al.
(författare)
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A mixed adaptive variational multiscale method with applications in oil reservoir simulation
- 2009
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Ingår i: Mathematical Models and Methods in Applied Sciences. - : World Scientific. - 0218-2025. ; 19:7, s. 1017-1042
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Tidskriftsartikel (refereegranskat)abstract
- We present a mixed adaptive variational multiscale method for solving elliptic second-order problems. This work is an extension of the adaptive variational multiscale method (AVMS), introduced by Larson and Malqvist,(15-17) to a mixed formulation. The method is based on a particular splitting into coarse and fine scales together with a systematic technique for approximation of the fine scale part based on solution of decoupled localized subgrid problems. We present the mixed AVMS method and derive a posteriori error estimates both for linear functionals and the energy norm. Based on the estimates we propose adaptive algorithms for automatic tuning of critical discretization parameters. Finally, we present numerical examples on a two-dimensional slice of an oil reservoir.
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- Larson, Mats G., et al.
(författare)
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A posteriori error estimates for mixed finite element approximations of elliptic problems
- 2008
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Ingår i: Numerische Mathematik. - : Springer. - 0029-599X .- 0945-3245. ; 108:3, s. 487-500
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Tidskriftsartikel (refereegranskat)abstract
- We derive residual based a posteriori error estimates of the flux in L 2-norm for a general class of mixed methods for elliptic problems. The estimate is applicable to standard mixed methods such as the Raviart–Thomas–Nedelec and Brezzi–Douglas–Marini elements, as well as stabilized methods such as the Galerkin-Least squares method. The element residual in the estimate employs an elementwise computable postprocessed approximation of the displacement which gives optimal order.
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- Larson, Mats G, et al.
(författare)
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An adaptive variational multiscale method for convection-diffusion problems
- 2009
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Ingår i: Communications in Numerical Methods in Engineering. - : Wiley-Blackwell. - 1069-8299 .- 1099-0887. ; 25:1, s. 65-79
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Tidskriftsartikel (refereegranskat)abstract
- The adaptive variational multiscale method is an extension of the variational multiscale method where the line-scale part of the solution is approximated by a sum of numerically computed solutions to localized subgrid problems. Furthermore, the crucial discretization parameters are chosen automatically by an adaptive algorithm based on a posteriori error estimates. This method has been developed for diffusion-dominated problems and applied to multiscale problems that arise in oil reservoir Simulation. In this paper, we extend the method to convection-diffusion problems. We present it duality based a posteriori error representation formula and an adaptive algorithm that tunes the fine-scale mesh size and the patch sizes of the local problems. Numerical results show rapid convergence of the adaptive algorithm. Copyright (c) 2008 John Wiley & Sons, Ltd.
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- Målqvist, Axel, 1978
(författare)
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Adaptive variational multiscale methods
- 2005
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis we present a new adaptive multiscale method for solving elliptic partial differential equations. The method is based on numerical solution of decoupled local fine scale problems on patches. Critical parameters such as fine and coarse scale mesh size and patch size are tuned automatically by an adaptive algorithm based on a posteriori error estimates. We extend the method to a mixed formulation of the Poisson equation and derive error estimates in this case as well. We also present a framework for adaptivity based on a posteriori error estimates for multi-physics problems. We study a coupled flow and transport problem and derive an a posteriori error estimate for a linear functional by introducing two dual problems, one associated with the transport equation and one associated with the flow equation. We also apply this method to a model problem in oil reservoir simulation.
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