| 1. |
- Jensen, M., et al.
(författare)
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Finite element convergence for the time-dependent Joule heating problem with mixed boundary conditions
- 2022
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Ingår i: Ima Journal of Numerical Analysis. - : Oxford University Press (OUP). - 0272-4979 .- 1464-3642. ; 42:1, s. 199-228
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Tidskriftsartikel (refereegranskat)abstract
- We prove strong convergence for a large class of finite element methods for the time-dependent Joule heating problem in three spatial dimensions with mixed boundary conditions on Lipschitz domains. We consider conforming subspaces for the spatial discretization and the backward Euler scheme for the temporal discretization. Furthermore, we prove uniqueness and higher regularity of the solution on creased domains and additional regularity in the interior of the domain. Due to a variational formulation with a cut-off functional, the convergence analysis does not require a discrete maximum principle, permitting approximation spaces suitable for adaptive mesh refinement, responding to the difference in regularity within the domain.
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| 2. |
- Görtz, Morgan, 1994, et al.
(författare)
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Multiscale methods for solving wave equations on spatial networks
- 2023
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825. ; 410
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Tidskriftsartikel (refereegranskat)abstract
- We present and analyze a multiscale method for wave propagation problems, posed on spatial networks. By introducing a coarse scale, using a finite element space interpolated onto the network, we construct a discrete multiscale space using the localized orthogonal decomposition (LOD) methodology. The spatial discretization is then combined with an energy conserving temporal scheme to form the proposed method. Under the assumption of well-prepared initial data, we derive an a priori error bound of optimal order with respect to the space and time discretization. In the analysis, we combine the theory derived for stationary elliptic problems on spatial networks with classical finite element results for hyperbolic problems. Finally, we present numerical experiments that confirm our theoretical findings. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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| 3. |
- Hellman, Fredrik, et al.
(författare)
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Numerical upscaling for heterogeneous materials in fractured domains
- 2021
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Ingår i: Esaim - Mathematical Modelling and Numerical Analysis - Modelisation Mathematique Et Analyse Numerique. - : EDP Sciences. - 0764-583X .- 2822-7840 .- 1290-3841. ; 55, s. S761-S784
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Tidskriftsartikel (refereegranskat)abstract
- We consider numerical solution of elliptic problems with heterogeneous diffusion coefficients containing thin highly conductive structures. Such problems arise e.g. in fractured porous media, reinforced materials, and electric circuits. The main computational challenge is the high resolution needed to resolve the data variation. We propose a multiscale method that models the thin structures as interfaces and incorporate heterogeneities in corrected shape functions. The construction results in an accurate upscaled representation of the system that can be used to solve for several forcing functions or to simulate evolution problems in an efficient way. By introducing a novel interpolation operator, defining the fine scale of the problem, we prove exponential decay of the shape functions which allows for a sparse approximation of the upscaled representation. An a priori error bound is also derived for the proposed method together with numerical examples that verify the theoretical findings. Finally we present a numerical example to show how the technique can be applied to evolution problems.
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| 4. |
- Ljung, Per, et al.
(författare)
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A generalized finite element method for the strongly damped wave equation with rapidly varying data
- 2021
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Ingår i: Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique Et Analyse Numerique. - : EDP Sciences. - 0764-583X .- 2822-7840 .- 1290-3841. ; 55:4, s. 1375-1403
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Tidskriftsartikel (refereegranskat)abstract
- We propose a generalized finite element method for the strongly damped wave equation with highly varying coefficients. The proposed method is based on the localized orthogonal decomposition introduced in Malqvist and Peterseim [Math. Comp. 83 (2014) 2583-2603], and is designed to handle independent variations in both the damping and the wave propagation speed respectively. The method does so by automatically correcting for the damping in the transient phase and for the propagation speed in the steady state phase. Convergence of optimal order is proven in L-2(H-1)-norm, independent of the derivatives of the coefficients. We present numerical examples that confirm the theoretical findings.
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| 5. |
- Hellman, Filip, 1984, et al.
(författare)
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Multiscale Mixed Finite Elements
- 2016
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Ingår i: Discrete and Continuous Dynamical Systems - Series S. - : American Institute of Mathematical Sciences (AIMS). - 1937-1632 .- 1937-1179. ; 9:5, s. 1269-1298
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Tidskriftsartikel (refereegranskat)abstract
- In this work, we propose a mixed finite element method for solving elliptic multiscale problems based on a localized orthogonal decomposition (LOD) of Raviart-Thomas finite element spaces. It requires to solve local problems in small patches around the elements of a coarse grid. These computations can be perfectly parallelized and are cheap to perform. Using the results of these patch problems, we construct a low dimensional multiscale mixed finite element space with very high approximation properties. This space can be used for solving the original saddle point problem in an efficient way. We prove convergence of our approach, independent of structural assumptions or scale separation. Finally, we demonstrate the applicability of our method by presenting a variety of numerical experiments, including a comparison with an MsFEM approach.
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| 6. |
- Elfverson, Daniel, et al.
(författare)
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Multiscale methods for problems with complex geometry
- 2017
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 321, s. 103-123
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Tidskriftsartikel (refereegranskat)abstract
- We propose a multiscale method for elliptic problems on complex domains, e.g. domains with cracks or complicated boundary. For local singularities this paper also offers a discrete alternative to enrichment techniques such as XFEM. We construct corrected coarse test and trail spaces which takes the fine scale features of the computational domain into account. The corrections only need to be computed in regions surrounding fine scale geometric features. We achieve linear convergence rate in the energy norm for the multiscale solution. Moreover, the conditioning of the resulting matrices is not affected by the way the domain boundary cuts the coarse elements in the background mesh. The analytical findings are verified in a series of numerical experiments.
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| 7. |
- Engwer, C., et al.
(författare)
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Efficient implementation of the localized orthogonal decomposition method
- 2019
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier B.V.. - 0045-7825 .- 1879-2138. ; 350, s. 123-153
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we present algorithms for an efficient implementation of the Localized Orthogonal Decomposition method (LOD). The LOD is a multiscale method for the numerical simulation of partial differential equations with a continuum of inseparable scales. We show how the method can be implemented in a fairly standard Finite Element framework and discuss its realization for different types of problems, such as linear elliptic problems with rough coefficients and linear eigenvalue problems.
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| 8. |
- Målqvist, Axel, 1978, et al.
(författare)
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Multiscale techniques for parabolic equations
- 2018
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Ingår i: Numerische Mathematik. - : Springer Science and Business Media LLC. - 0029-599X .- 0945-3245. ; 138:1, s. 191-217
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Tidskriftsartikel (refereegranskat)abstract
- We use the local orthogonal decomposition technique introduced in MAlqvist and Peterseim (Math Comput 83(290):2583-2603, 2014) to derive a generalized finite element method for linear and semilinear parabolic equations with spatial multiscale coefficients. We consider nonsmooth initial data and a backward Euler scheme for the temporal discretization. Optimal order convergence rate, depending only on the contrast, but not on the variations of the coefficients, is proven in the -norm. We present numerical examples, which confirm our theoretical findings.
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| 9. |
- 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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| 10. |
- 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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