| 1. |
- Brehier, Charles-Edouard, et al.
(författare)
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Analysis of a positivity-preserving splitting scheme for some semilinear stochastic heat equations
- 2024
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Ingår i: ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS. - 2822-7840 .- 2804-7214. ; 58:4, s. 1317-1346
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Tidskriftsartikel (refereegranskat)abstract
- We construct a positivity-preserving Lie-Trotter splitting scheme with finite difference discretization in space for approximating the solutions to a class of semilinear stochastic heat equations with multiplicative space-time white noise. We prove that this explicit numerical scheme converges in the mean-square sense, with rate 1/4 in time and rate 1/2 in space, under appropriate CFL conditions. Numerical experiments illustrate the superiority of the proposed numerical scheme compared with standard numerical methods which do not preserve positivity.
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| 2. |
- Huang, Xin, et al.
(författare)
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Canonical mean-field molecular dynamics derived from quantum mechanics
- 2022
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Ingår i: ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS. - : EDP Sciences. - 2822-7840 .- 2804-7214. ; 56:6, s. 2197-2238
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Tidskriftsartikel (refereegranskat)abstract
- Canonical quantum correlation observables can be approximated by classical molecular dynamics. In the case of low temperature the ab initio molecular dynamics potential energy is based on the ground state electron eigenvalue problem and the accuracy has been proven to be O(M-1), provided the first electron eigenvalue gap is sufficiently large compared to the given temperature and M is the ratio of nuclei and electron masses. For higher temperature eigenvalues corresponding to excited electron states are required to obtain O(M-1) accuracy and the derivations assume that all electron eigenvalues are separated, which for instance excludes conical intersections. This work studies a mean-field molecular dynamics approximation where the mean-field Hamiltonian for the nuclei is the partial trace h := Tr(He-beta H)/Tr(e(-beta H)) with respect to the electron degrees of freedom and H is the Weyl symbol corresponding to a quantum many body Hamiltonian (sic). It is proved that the mean-field molecular dynamics approximates canonical quantum correlation observables with accuracy O(M-1 + t epsilon(2)), for correlation time t where epsilon(2) is related to the variance of mean value approximation h. Furthermore, the proof derives a precise asymptotic representation of the Weyl symbol of the Gibbs density operator using a path integral formulation. Numerical experiments on a model problem with one nuclei and two electron states show that the mean-field dynamics has similar or better accuracy than standard molecular dynamics based on the ground state electron eigenvalue.
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| 3. |
- 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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| 4. |
- Romano, Luigi, 1994-, et al.
(författare)
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Finite element modelling of linear rolling contact problems
- 2024
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Ingår i: ESAIM: Mathematical Modelling and Numerical Analysis. - 2822-7840 .- 2804-7214.
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Tidskriftsartikel (refereegranskat)abstract
- The present work is devoted to the finite element modelling of linear hyperbolic rolling contact problems. The main equations encountered in rolling contact mechanics are reviewed in the first part of the paper, with particular emphasis on applications from automotive and vehicle engineering. In contrast to the common hyperbolic systems found in the literature, such equations include integral and boundary terms, as well as time-varying transport velocities, that require special treatment. In this context, existence and uniqueness properties are discussed within the theoretical framework offered by the semigroup theory. The second part of the paper is then dedicated to recovering approximated solutions to the considered problems, by combining discontinuous Galerkin finite element methods (DGMs) with explicit Runge-Kutta (RK) schemes of the first and second order for time discretisation. Under opportune assumptions on the smoothness of the sought solutions, and owing to certain generalised Courant-Friedrichs-Lewy (CFL) conditions, quasi-optimal error bounds are derived for the complete discrete schemes. The proposed algorithms are then tested on simple scalar equations in one space dimension. Numerical experiments seem to suggest the theoretical error estimates to be sharp.
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| 5. |
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| 6. |
- Bürger, Raimund, et al.
(författare)
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Numerical schemes for a moving-boundary convection-diffusion-reaction model of sequencing batch reactors
- 2023
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Ingår i: ESAIM: Mathematical Modelling and Numerical Analysis. - 2822-7840. ; 57:5, s. 2931-2976
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Tidskriftsartikel (refereegranskat)abstract
- Sequencing batch reactors (SBRs) are devices widely used in wastewater treatment, chemical engineering, and other areas. They allow for the sedimentation and compression of solid particles of biomass simultaneously with biochemical reactions with nutrients dissolved in the liquid. The kinetics of these reactions may be given by one of the established activated sludge models (ASMx). An SBR is operated in various stages and is equipped with a movable extraction and fill device and a discharge opening. A one-dimensional model of this unit can be formulated as a moving-boundary problem for a degenerating system of convection-diffusion-reaction equations whose unknowns are the concentrations of the components forming the solid and liquid phases, respectively. This model is transformed to a fixed computational domain and is discretized by an explicit monotone scheme along with an alternative semi-implicit variant. The semi-implicit variant is based on solving, during each time step, a system of nonlinear equations for the total solids concentration followed by the solution of linear systems for the solid component percentages and liquid component concentrations. It is demonstrated that the semi-implicit scheme is well posed and that both variants produce approximations that satisfy an invariant region principle: solids concentrations are nonnegative and less or equal to a set maximal one, percentages are nonnegative and sum up to one, and substrate concentrations are nonnegative. These properties are achieved under a Courant-Friedrichs-Lewy (CFL) condition that is less restrictive for the semi-implicit than for the explicit variant. Numerical examples with realistic parameters illustrate that as a consequence, the semi-implicit variant is more efficient than the explicit one.
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| 7. |
- Caubet, Fabien, et al.
(författare)
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New Transmission Condition Accounting For Diffusion Anisotropy In Thin Layers Applied To Diffusion MRI
- 2017
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Ingår i: ESAIM: M2AN. - : EDP Sciences. - 2822-7840. ; 51, s. 1279-1301
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Tidskriftsartikel (refereegranskat)abstract
- The Bloch-Torrey Partial Differential Equation (PDE) can be used to model the diffusion Magnetic Resonance Imaging (dMRI) signal in biological tissue. In this paper, we derive an Anisotropic Diffusion Transmission Condition (ADTC) for the Bloch-Torrey PDE that accounts for anisotropic diffusion inside thin layers. Such diffusion occurs, for example, in the myelin sheath surrounding the axons of neurons. This ADTC can be interpreted as an asymptotic model of order two with respect to the layer thickness and accounts for water diffusion in the normal direction that is low compared to the tangential direction. We prove the uniform stability of the asymptotic model with respect to the layer thickness and a mass conservation property. We also prove the theoretical quadratic accuracy of the ADTC. Finally, numerical tests validate these results and show that our model gives a better approximation of the dMRI signal than a simple transmission condition that assumes isotropic diffusion in the layers.
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| 8. |
- Cohen, David, et al.
(författare)
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High order numerical methods for highly oscillatory problem
- 2015
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Ingår i: Mathematical Modelling and Numerical Analysis. - : EDP Sciences. - 0764-583X .- 1290-3841 .- 2822-7840. ; 49:3, s. 695-711
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Tidskriftsartikel (refereegranskat)abstract
- This paper is concerned with the numerical solution of nonlinear Hamiltonian highly oscillatory systems of second-order differential equations of a special form. We present numerical methods of high asymptotic as well as time stepping order based on the modulated Fourier expansion of the exact solution. In particular we obtain time stepping orders higher than 2 with only a finite energy assumption on the initial values of the problem. In addition, the stepsize of these new numerical integrators is not restricted by the high frequency of the problem. Furthermore, numerical experiments on the modified Fermi–Pasta–Ulam problem as well as on a one dimensional model of a diatomic gas with short-range interaction forces support our investigations.
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| 9. |
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| 10. |
- 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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| 11. |
- Henning, Patrick, et al.
(författare)
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A localized orthogonal decomposition method for semi-linear elliptic problems
- 2014
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Ingår i: Mathematical Modelling and Numerical Analysis. - : EDP Sciences. - 2822-7840 .- 0764-583X .- 1290-3841. ; 48:5, s. 1331-1349
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we propose and analyze a localized orthogonal decomposition (LOD) method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. This Galerkin-type method is based on a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations on small patches that have a diameter of order H | log (H) | where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coefficients, we give a rigorous proof for a linear convergence of the H1-error with respect to the coarse mesh size even for rough coefficients. To solve the corresponding system of algebraic equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space.
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| 12. |
- Kirchner, Kristin, 1987, et al.
(författare)
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Maxwell's equations for conductors with impedance boundary conditions: Discontinuous Galerkin and Reduced Basis Methods
- 2016
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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. ; 50:6, s. 1763-1787
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Tidskriftsartikel (refereegranskat)abstract
- We consider Maxwell's equations with impedance boundary conditions on a conductive polyhedron with polyhedral holes. Well-posedness of the variational formulation is proven, a hp-discontinuous Galerkin (hp-dG) approximation as well as a priori error estimates are introduced. Next, we use the frequency. as a parameter in a multi-query context. For this purpose, we derive a Reduced Basis Method (RBM) based upon the dG formulation as well as the corresponding a posteriori error bound. Numerical results indicate the efficiency and the robustness of the scheme.
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| 13. |
- Kovacs, Mihaly, 1977, et al.
(författare)
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Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise
- 2020
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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. ; 54:6, s. 2199-2227
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Tidskriftsartikel (refereegranskat)abstract
- The numerical approximation of the mild solution to a semilinear stochastic wave equation driven by additive noise is considered. A standard finite element method is employed for the spatial approximation and a a rational approximation of the exponential function for the temporal approximation. First, strong convergence of this approximation in both positive and negative order norms is proven. With the help of Malliavin calculus techniques this result is then used to deduce weak convergence rates for the class of twice continuously differentiable test functions with polynomially bounded derivatives. Under appropriate assumptions on the parameters of the equation, the weak rate is found to be essentially twice the strong rate. This extends earlier work by one of the authors to the semilinear setting. Numerical simulations illustrate the theoretical results.
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| 14. |
- 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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| 15. |
- Målqvist, Axel, 1978, et al.
(författare)
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A Generalized Finite Element Method for Linear Thermoelasticity
- 2017
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Ingår i: Mathematical Modelling and Numerical Analysis. - : EDP Sciences. - 2822-7840 .- 0764-583X .- 1290-3841. ; 51:4, s. 1145-1171
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Tidskriftsartikel (refereegranskat)abstract
- We propose and analyze a generalized finite element method designed for linear quasistatic thermoelastic systems with spatial multiscale coefficients. The method is based on the local orthogonal decomposition technique introduced by Malqvist and Peterseim (Math. Comp. 83 (2014) 2583-2603). We prove convergence of optimal order, independent of the derivatives of the coefficients, in the spatial H-1-norm. The theoretical results are confirmed by numerical examples.
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| 16. |
- Målqvist, Axel, 1978, et al.
(författare)
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Generalized finite element methods for quadratic eigenvalue problems
- 2017
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Ingår i: ESAIM: Mathematical Modelling and Numerical Analysis. - : EDP Sciences. - 0764-583X .- 1290-3841 .- 2822-7840. ; 51:1, s. 147-163
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Tidskriftsartikel (refereegranskat)abstract
- © EDP Sciences, SMAI 2016. We consider a large-scale quadratic eigenvalue problem (QEP), formulated using P1 finite elements on a fine scale reference mesh. This model describes damped vibrations in a structural mechanical system. In particular we focus on problems with rapid material data variation, e.g., composite materials. We construct a low dimensional generalized finite element (GFE) space based on the localized orthogonal decomposition (LOD) technique. The construction involves the (parallel) solution of independent localized linear Poisson-type problems. The GFE space is used to compress the large-scale algebraic QEP to a much smaller one with a similar modeling accuracy. The small scale QEP can then be solved by standard techniques at a significantly reduced computational cost. We prove convergence with rate for the proposed method and numerical experiments confirm our theoretical findings.
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| 17. |
- Thomee, Vidar, 1933, et al.
(författare)
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Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations
- 2006
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Ingår i: M2AN Math. Model. Numer. Anal.. - : EDP Sciences. - 0764-583X .- 2822-7840 .- 1290-3841. ; 40:5, s. 923-937
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Tidskriftsartikel (refereegranskat)abstract
- In recent years several papers have been devoted to stability and smoothing properties in maximum-norm of finite element discretizations of parabolic problems. Using the theory of analytic semigroups it has been possible to rephrase such properties as bounds for the resolvent of the associated discrete elliptic operator. In all these cases the triangulations of the spatial domain has been assumed to be quasiuniform. In the present paper we show a resolvent estimate, in one and two space dimensions, under weaker conditions on the triangulations than quasiuniformity. In the two-dimensional case, the bound for the resolvent contains a logarithmic factor.
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