arXiv subjects

Barbara Verfürth

Publications and source records attributed to Barbara Verfürth.

At least 19 recordsLinked to original sources

arXiv2026

Interface Conditions for Wave Propagation Through a Time-varying Metasurface

We study wave propagation through a time-modulated thin heterogeneous layer. The layer is assumed to have a thickness of order $\es\ll 1$ and its material properties exhibit rapid oscillations on multiple spatial and temporal scales. We aim to rigorously derive the effective model and the corresponding interface conditions for wave propagation through the limiting interface. To perform the homogenization together with dimension reduction, we generalize the notion of two-scale convergence for thin layer introduced by Neuss-Radu and Jäger (2007) to a multiple space-time scale framework. One of the main analytical difficulties arising in the homogenization analysis is that, in general, a uniform energy estimate cannot be obtained for a wave equation with time-varying coefficients. Therefore, we identify two physically relevant classes of coefficients for which we can derive a uniform energy bound. These include coefficients with traveling wave-type modulations of their properties. Using the energy estimates and the multiscale convergence for thin layer concept, we derive the effective model, which consists of linear wave equations in the bulk domains coupled through a nonstandard jump condition at the interface. This jump condition is governed by a wave-type dynamical equation with effective macroscopic coefficients determined by suitable cell problems of elliptic and hyperbolic type. Finally, we discuss the uniqueness of the effective model.

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arXiv2026

Optimization-Based Identification of Effective Coefficients for Wave Equations in Spatio-Temporal Metamaterials

We study the identification of effective coefficients for wave equations in heterogeneous media. Such equations arise in the modeling of spatio-temporal metamaterials, where the underlying material properties exhibit variations in both space and time. While homogenization provides effective models in the asymptotic regime of vanishing microscopic scales, determining macroscopic parameters from observations of wave propagation remains challenging. We introduce an optimization-based method that identifies a constant effective coefficient by minimizing a cost functional. The approach is designed for situations in which the underlying space-time-dependent coefficient is unknown, while the solution is available in space and time by measurements. We extend optimization-based coefficient identification techniques from elliptic multiscale problems to wave equations and prove that, provided a homogenized limit exists, the identified coefficient converges to the homogenized coefficient as the microscopic scale tends to zero. Furthermore, we establish convergence of the corresponding effective solution towards the heterogeneous solution, including strong convergence in $L^2$ and weak convergence of first-order space and time derivatives. Numerical experiments further demonstrate the performance of the method for non-periodic space-time heterogeneous media, including cases for which a homogenized limit is not known to exist.

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arXiv2026

Subspace decomposition with defect diffusion coefficient

Elliptic diffusion problems with multiscale heterogeneous coefficients lead to poorly conditioned discrete systems and therefore require effective preconditioning strategies. While subspace decomposition preconditioners perform well for fixed realizations of the coefficient, their repeated construction becomes prohibitively expensive in uncertainty quantification settings, particularly in Monte-Carlo simulations, where a large number of fine-scale realizations must be treated. In this study, we propose an offline-online approximation of a subspace decomposition preconditioner that exploits the localized structure of the random defects. The preconditioner is constructed from local subspace solves that are precomputed offline for a small set of reference configurations and efficiently combined online for arbitrary realizations. We analyze the spectral properties of the resulting offline-online approximation operator and confirm its robustness and efficiency through numerical experiments.

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arXiv2026

Adaptive Iterative Numerical Homogenization for Quasilinear Nonmonotone Elliptic PDE

We propose and analyze an adaptive iterative numerical homogenization method to approximate the solution of a class of quasilinear nonmonotone elliptic problems that is of multiscale nature. The method is based on the technique of the Localized Orthogonal Decomposition (LOD) applied to the linear problems in each step of a Kačanov iteration. In this approach, the multiscale basis is recomputed adaptively in each iteration and a linear problem is solved with this updated multiscale space, where in both steps the nonlinearity is evaluated using the approximation of the previous iteration. As a key component of the proposed approach, we present a locally computable error indicator, which at each iteration identifies the basis functions requiring updates, while previously computed basis functions are retained whenever possible. We provide a priori error estimates and show convergence of the method, requiring only higher integrability of the right-hand side, but no higher differentiability of the solution itself. Furthermore, we discuss how to adapt the proposed adaptive iterative LOD in the context of Newton's method as iteration scheme. Numerical experiments illustrate the theory and validate the applicability of the proposed method.

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arXiv2026

Local and nonlocal homogenization of wave propagation in time-varying media

Temporal metamaterials are artificially manufactured materials with time-dependent material properties that exhibit interesting phenomena when waves propagate through them. The propagation of electromagnetic waves in such time-varying dielectric media is governed by Maxwell's equations, which lead to wave equations with temporal highly oscillatory coefficients for the electric and magnetic fields. In this study, we analyze the effective behavior of electromagnetic fields in time-varying metamaterials using a formal two-scale asymptotic expansion. We provide a mathematical derivation of the effective equations for the leading-order homogenized solution, as well as for the first- and second-order corrections of the effective solution. While the effective solution and the first-order correction are governed by local material laws, we reveal a nonlocal constitutive relation for the second-order corrections. Special attention is also paid to temporal interface conditions through initial values of the homogenized equations. The results provide a mathematically justified framework for the effective description of wave-type equations of time-varying media, applicable to models in optics, elasticity, and acoustics.

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arXiv2025

Linearized Localized Orthogonal Decomposition for Quasilinear Nonmonotone Elliptic PDE

In this paper, we propose and analyze a multiscale method for a class of quasilinear elliptic problems of nonmonotone type with spatially multiscale coefficient. The numerical approach is inspired by the Localized Orthogonal Decomposition (LOD), so that we do not require structural assumptions such as periodicity or scale separation and only need minimal regularity assumptions on the coefficient.To construct the multiscale space, we solve linear fine-scale problems on small local subdomains, for which we consider two different linearization techniques. For both, we present a rigorous well-posedness analysis and convergence estimates in the $H^1$-semi norm. We compare and discuss theoretically and numerically the performance of our strategies for different linearization points. Numerical experiments underline the theoretical findings and illustrate the applicability of the method.

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arXiv2025

Offline-online approximation of multiscale eigenvalue problems with random defects

In this paper, we consider an elliptic eigenvalue problem with multiscale, randomly perturbed coefficients. For an efficient and accurate approximation of the solutions for many different realizations of the coefficient, we propose a computational multiscale method in the spirit of the Localized Orthogonal Decomposition (LOD) method together with an offline-online strategy similar to [Målqvist, Verfürth, ESIAM Math. Model. Numer. Anal., 56(1):237-260, 2022]. The offline phase computes and stores local contributions to the LOD stiffness matrix for selected defect configurations. Given any perturbed coefficient, the online phase combines the pre-computed quantities in an efficient manner. We further propose a modification in the online phase, for which numerical results indicate enhanced performances for moderate and high defect probabilities. We show rigorous a priori error estimates for eigenfunctions as well as eigenvalues.

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arXiv2024

Error analysis of an implicit-explicit time discretization scheme for semilinear wave equations with application to multiscale problems

We present an implicit-explicit (IMEX) scheme for semilinear wave equations with strong damping. By treating the nonlinear, nonstiff term explicitly and the linear, stiff part implicitly, we obtain a method which is not only unconditionally stable but also highly efficient. Our main results are error bounds of the full discretization in space and time for the IMEX scheme combined with a general abstract space discretization. As an application, we consider the heterogeneous multiscale method for wave equations with highly oscillating coefficients in space for which we show spatial and temporal convergence rates by using the abstract result.

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arXiv2023

Statistical Variational Data Assimilation

This paper is a contribution in the context of variational data assimilation combined with statistical learning. The framework of data assimilation traditionally uses data collected at sensor locations in order to bring corrections to a numerical model designed using knowledge of the physical system of interest. However, some applications do not have available data at all times, but only during an initial training phase. Hence, we suggest to combine data assimilation with statistical learning methods; namely, deep learning. More precisely, for time steps at which data is unavailable, a surrogate deep learning model runs predictions of the `true' data which is then assimilated by the new model. In this paper, we also derive a priori error estimates on this SVDA approximation. Finally, we assess the method by numerical test cases.

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arXiv2023

Wave propagation in high-contrast media: periodic and beyond

This work is concerned with the classical wave equation with a high-contrast coefficient in the spatial derivative operator. We first treat the periodic case, where we derive a new limit in the one-dimensional case. The behavior is illustrated numerically and contrasted to the higher-dimensional case. For general unstructured high-contrast coefficients, we present the Localized Orthogonal Decomposition and show a priori error estimates in suitably weighted norms. Numerical experiments illustrate the convergence rates in various settings.

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arXiv2023

Higher-order finite element methods for the nonlinear Helmholtz equation

In this work, we analyze the finite element method with arbitrary but fixed polynomial degree for the nonlinear Helmholtz equation with impedance boundary conditions. We show well-posedness and error estimates of the finite element solution under a resolution condition between the wave number $k$, the mesh size $h$ and the polynomial degree $p$ of the form ``$k(kh)^{p+1}$ sufficiently small'' and a so-called smallness of the data assumption. For the latter, we prove that the logarithmic dependence in $h$ from the case $p=1$ in [H.~Wu, J.~Zou, \emph{SIAM J.~Numer.~Anal.} 56(3): 1338-1359, 2018] can be removed for $p\geq 2$. We show convergence of two different fixed-point iteration schemes. Numerical experiments illustrate our theoretical results and compare the robustness of the iteration schemes with respect to the size of the nonlinearity and the right-hand side data.

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arXiv2022

Fully discrete Heterogeneous Multiscale Method for parabolic problems with multiple spatial and temporal scales

The aim of this work is the numerical homogenization of a parabolic problem with several time and spatial scales using the heterogeneous multiscale method. We replace the actual cell problem with an alternate one, using Dirichlet boundary and initial values instead of periodic boundary and time conditions. Further, we give a detailed a priori error analysis of the fully discretized, i.e., in space and time for both the macroscopic and the cell problem, method. Numerical experiments illustrate the theoretical convergence rates.

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arXiv2021

Nonlinear Helmholtz equations with sign-changing diffusion coefficient

In this paper we study nonlinear Helmholtz equations with sign-changing diffusion coefficients on bounded domains. The existence of an orthonormal basis of eigenfunctions is established making use of weak T-coercivity theory. All eigenvalues are proved to be bifurcation points and the bifurcating branches are investigated both theoretically and numerically. In a one-dimensional model example we obtain the existence of infinitely many bifurcating branches that are mutually disjoint, unbounded, and consist of solutions with a fixed nodal pattern.

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arXiv2021

Multiscale Scattering in Nonlinear Kerr-Type Media

We propose a multiscale approach for a nonlinear Helmholtz problem with possible oscillations in the Kerr coefficient, the refractive index, and the diffusion coefficient. The method does not rely on structural assumptions on the coefficients and combines the multiscale technique known as Localized Orthogonal Decomposition with an adaptive iterative approximation of the nonlinearity. We rigorously analyze the method in terms of well-posedness and convergence properties based on suitable assumptions on the initial data and the discretization parameters. Numerical examples illustrate the theoretical error estimates and underline the practicability of the approach.

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arXiv2021

An offline-online strategy for multiscale problems with random defects

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 coarse-scale 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.

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arXiv2021

Numerical upscaling for wave equations with time-dependent multiscale coefficients

In this paper, we consider the classical wave equation with time-dependent, spatially multiscale coefficients. We propose a fully discrete computational multiscale method in the spirit of the localized orthogonal decomposition in space with a backward Euler scheme in time. We show optimal convergence rates in space and time beyond the assumptions of spatial periodicity or scale separation of the coefficients. Further, we propose an adaptive update strategy for the time-dependent multiscale basis. Numerical experiments illustrate the theoretical results and showcase the practicability of the adaptive update strategy.

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arXiv2020

Numerical homogenization for nonlinear strongly monotone problems

In this work we introduce and analyze a new multiscale method for strongly nonlinear monotone equations in the spirit of the Localized Orthogonal Decomposition. A problem-adapted multiscale space is constructed by solving linear local fine-scale problems which is then used in a generalized finite element method. The linearity of the fine-scale problems allows their localization and, moreover, makes the method very efficient to use. The new method gives optimal a priori error estimates up to linearization errors. The results neither require structural assumptions on the coefficient such as periodicity or scale separation nor higher regularity of the solution. The effect of different linearization strategies is discussed in theory and practice. Several numerical examples including stationary Richards equation confirm the theory and underline the applicability of the method.

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arXiv2020

A multiscale method for heterogeneous bulk-surface coupling

In this paper, we construct and analyze a multiscale (finite element) method for parabolic problems with heterogeneous dynamic boundary conditions. As origin, we consider a reformulation of the system in order to decouple the discretization of bulk and surface dynamics. This allows us to combine multiscale methods on the boundary with standard Lagrangian schemes in the interior. We prove convergence and quantify explicit rates for low-regularity solutions, independent of the oscillatory behavior of the heterogeneities. As a result, coarse discretization parameters, which do not resolve the fine scales, can be considered. The theoretical findings are justified by a number of numerical experiments including dynamic boundary conditions with random diffusion coefficients.

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