| 1. |
- 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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| 2. |
- 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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| 3. |
- 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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| 4. |
- 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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| 5. |
- 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
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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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| 6. |
- Lang, Annika, 1980, et al.
(författare)
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LOCALIZED ORTHOGONAL DECOMPOSITION FOR A MULTISCALE PARABOLIC STOCHASTIC PARTIAL DIFFERENTIAL EQUATION
- 2024
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Ingår i: MULTISCALE MODELING & SIMULATION. - 1540-3459 .- 1540-3467. ; 22:1, s. 204-229
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Tidskriftsartikel (refereegranskat)abstract
- A multiscale method is proposed for a parabolic stochastic partial differential equation with additive noise and highly oscillatory diffusion. The framework is based on the localized orthogonal decomposition (LOD) method and computes a coarse -scale representation of the elliptic operator, enriched by fine -scale information on the diffusion. Optimal order strong convergence is derived. The LOD technique is combined with a (multilevel) Monte Carlo estimator and the weak error is analyzed. Numerical examples that confirm the theoretical findings are provided, and the computational efficiency of the method is highlighted.
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| 7. |
- 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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| 8. |
- Målqvist, Axel, 1978, et al.
(författare)
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An offline-online strategy for multiscale problems with random defects
- 2022
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Ingår i: ESAIM: Mathematical Modelling and Numerical Analysis. - : EDP Sciences. - 0764-583X .- 2822-7840 .- 2804-7214. ; 56:1, s. 237-260
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we propose an offline-online strategy based on the Localized Orthogonal Decomposition (LOD) method for elliptic multiscale problems with randomly perturbed diffusion coefficient. We consider a periodic deterministic coefficient with local defects that occur with probability p. The offline phase pre-computes entries to global LOD stiffness matrices on a single reference element (exploiting the periodicity) for a selection of defect configurations. Given a sample of the perturbed diffusion the corresponding LOD stiffness matrix is then computed by taking linear combinations of the pre-computed entries, in the online phase. Our computable error estimates show that this yields a good approximation of the solution for small p, which is illustrated by extensive numerical experiments. This makes the proposed technique attractive already for moderate sample sizes in a Monte Carlo simulation. © The authors. Published by EDP Sciences, SMAI 2022.
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| 9. |
- Edelvik, Fredrik, 1972, et al.
(författare)
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Numerical homogenization of spatial network models
- 2024
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - 0045-7825 .- 1879-2138. ; 418
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Tidskriftsartikel (refereegranskat)abstract
- We present and analyze a methodology for numerical homogenization of spatial networks models, e.g. heat conduction and linear deformation in large networks of slender objects, such as paper fibers. The aim is to construct a coarse model of the problem that maintains high accuracy also on the micro-scale. By solving decoupled problems on local subgraphs we construct a low dimensional subspace of the solution space with good approximation properties. The coarse model of the network is expressed by a Galerkin formulation and can be used to perform simulations with different source and boundary data, at a low computational cost. We prove optimal convergence to the micro-scale solution of the proposed method under mild assumptions on the homogeneity, connectivity, and locality of the network on the coarse scale. The theoretical findings are numerically confirmed for both scalar-valued (heat conduction) and vector-valued (linear deformation) models.
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| 10. |
- Elfverson, Daniel, et al.
(författare)
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A Multilevel Monte Carlo Method for Computing Failure Probabilities
- 2016
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Ingår i: SIAM-ASA Journal on Uncertainty Quantification. - : Society for Industrial & Applied Mathematics (SIAM). - 2166-2525. ; 4:1, s. 312-330
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Tidskriftsartikel (refereegranskat)abstract
- We propose and analyze a method for computing failure probabilities of systems modeled as numerical deterministic models (e.g., PDEs) with uncertain input data. A failure occurs when a functional of the solution to the model is below (or above) some critical value. By combining recent results on quantile estimation and the multilevel Monte Carlo method, we develop a method that reduces computational cost without loss of accuracy. We show how the computational cost of the method relates to error tolerance of the failure probability. For a wide and common class of problems, the computational cost is asymptotically proportional to solving a single accurate realization of the numerical model, i.e., independent of the number of samples. Significant reductions in computational cost are also observed in numerical experiments.
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