arXiv subjects

Anna Persson

Publications and source records attributed to Anna Persson.

11 recordsLinked to original sources

arXiv2026

Tangent-Space Multiscale Manifold Methods for Nonlinear Elliptic Problems

We introduce a tangent-space multiscale manifold method for nonlinear heterogeneous elliptic problems. The method represents the fine-scale solution by a nonlinear reconstruction of a coarse state. The ideal reconstruction eliminates fine scales through a constrained variational problem, and the computable reconstruction approximates this map by localized nonlinear patch solves blended with a partition of unity. Because the approximation set is a nonlinear manifold, the coarse equation is posed with tangent multiscale test functions. We also formulate a network-interpolated variant in which only the restricted patch outputs used by the partition-of-unity blend, together with their tangent actions, are approximated by local learned maps. For heterogeneous monotone nonlinear diffusion, we record the structural monotonicity, differentiability, patch-map regularity, and conditional perturbation estimates that separate the geometric stability mechanism from localization, residual, and optional learning defects. A rigorous a priori theory for the decay of the localization defect, and the resulting convergence rates in the coarse mesh size, is deferred to a separate analysis; here these defects are controlled conditionally and their decay is demonstrated numerically.

math.NA↗

arXiv2026

Numerical vortex resolution for the Gross-Pitaevskii equation in the rapid rotation Thomas-Fermi scaling

In this paper we analyze finite element approximations of ground states of the Gross-Pitaevskii equation in the rapid rotation Thomas-Fermi scaling. In this regime, the healing length and vortex core size are of order $\eps \ll 1$, while the effective confinement potential may degenerate as the angular velocity approaches a critical value. In this setting, we analyze the $\eps$-dependence of the ground states and show that the local flatness of the energy landscape plays a decisive role for numerical resolution. More precisely, we establish mesh size conditions that guarantee the existence of discrete ground states in finite element spaces which are quasi-best approximations of an exact ground state. In particular, we prove that the absolute $H^1$-error behaves asymptotically like $h/\eps^2$. However, to enter this asymptotic regime, the mesh size must satisfy a significantly stronger resolution condition than the natural requirement $h \lesssim \eps$. The additional restriction is governed by the first spectral gap of the Riemannian Hessian of the energy functional at the ground state, which measures the local flatness of the energy surface. With this, our results provide an explanation of the mesh resolution required to capture vortex structures in rapidly rotating Bose-Einstein condensates and highlight the interplay between vortex core size, spectral stability, and discretization accuracy.

math.NA↗

arXiv2024

Learning Nonlinear Finite Element Solution Operators using Multilayer Perceptrons and Energy Minimization

We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations (PDEs). The approach is based on a finite element discretization and aims at representing the solution operator by a multilayer perceptron (MLP) that takes problem data variables as input and gives a prediction of the finite element solution as output. The variables will typically correspond to parameters in a parametrization of input data such as boundary conditions, coefficients, and right-hand sides. The output will be an approximation of the corresponding finite element solution, thus enabling support and enhancement by the standard finite element method (FEM) both theoretically and practically. The loss function is most often an energy functional and we formulate efficient parallelizable training algorithms based on assembling the energy locally on each element. For large problems, the learning process can be made more efficient by using only a small fraction of randomly chosen elements in the mesh in each iteration. The approach is evaluated on several relevant test cases, where learning the finite element solution operator turns out to be beneficial, both in its own right but also by combination with standard FEM theory and software.

cs.LG↗

arXiv2021

On optimal convergence rates for discrete minimizers of the Gross-Pitaevskii energy in LOD spaces

In this paper we revisit a two-level discretization based on the Localized Orthogonal Decomposition (LOD). It was originally proposed in [P.Henning, A.M{\aa}lqvist, D.Peterseim. SIAM J. Numer. Anal.52-4:1525-1550, 2014] to compute ground states of Bose-Einstein condensates by finding discrete minimizers of the Gross-Pitaevskii energy functional. The established convergence rates for the method appeared however suboptimal compared to numerical observations and a proof of optimal rates in this setting remained open. In this paper we shall close this gap by proving optimal order error estimates for the $L^2$- and $H^1$-error between the exact ground state and discrete minimizers, as well as error estimates for the ground state energy and the ground state eigenvalue. In particular, the achieved convergence rates for the energy and the eigenvalue are of $6$th order with respect to the mesh size on which the discrete LOD space is based, without making any additional regularity assumptions. These high rates justify the use of very coarse meshes, which significantly reduces the computational effort for finding accurate approximations of ground states. In addition, we include numerical experiments that confirm the optimality of the new theoretical convergence rates, both for smooth and discontinuous potentials.

math.NA↗

arXiv2020

A generalized finite element method for the strongly damped wave equation with rapidly varying data

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 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.

math.NA↗

arXiv2019

Computational homogenization of time-harmonic Maxwell's equations

In this paper we consider a numerical homogenization technique for curl-curl-problems that is based on the framework of the Localized Orthogonal Decomposition and which was proposed in [D. Gallistl, P. Henning, B. Verf\"urth. SIAM J. Numer. Anal. 56-3:1570-1596, 2018] for problems with essential boundary conditions. The findings of the aforementioned work establish quantitative homogenization results for the time-harmonic Maxwell's equations that hold beyond assumptions of periodicity, however, a practical realization of the approach was left open. In this paper, we transfer the findings from essential boundary conditions to natural boundary conditions and we demonstrate that the approach yields a computable numerical method. We also investigate how boundary values of the source term can effect the computational complexity and accuracy. Our findings will be supported by various numerical experiments, both in $2D$ and $3D$.

math.NA↗

arXiv2018

Finite element convergence for the time-dependent Joule heating problem with mixed boundary conditions

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 the difference in regularity within the domain.

math.NA↗

arXiv2017

Multiscale differential Riccati equations for linear quadratic regulator problems

We consider approximations to the solutions of differential Riccati equations in the context of linear quadratic regulator problems, where the state equation is governed by a multiscale operator. Similarly to elliptic and parabolic problems, standard finite element discretizations perform poorly in this setting unless the grid resolves the fine-scale features of the problem. This results in unfeasible amounts of computation and high memory requirements. In this paper, we demonstrate how the localized orthogonal decomposition method may be used to acquire accurate results also for coarse discretizations, at the low cost of solving a series of small, localized elliptic problems. We prove second-order convergence (except for a logarithmic factor) in the $L^2$ operator norm, and first-order convergence in the corresponding energy norm. These results are both independent of the multiscale variations in the state equation. In addition, we provide a detailed derivation of the fully discrete matrix-valued equations, and show how they can be handled in a low-rank setting for large-scale computations. In connection to this, we also show how to efficiently compute the relevant operator-norm errors. Finally, our theoretical results are validated by several numerical experiments.

math.NA↗

arXiv2016

A generalized finite element method for linear thermoelasticity

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 M{\aa}lqvist and Peterseim (Math. Comp., 83(290): 2583--2603, 2014). 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.

math.NA↗

arXiv2016

A multiscale method for linear elasticity reducing Poisson locking

We propose a generalized finite element method for linear elasticity equations with highly varying and oscillating coefficients. The method is formulated in the framework of localized orthogonal decomposition techniques introduced by M{\aa}lqvist and Peterseim (Math. Comp., 83(290): 2583--2603, 2014). Assuming only $L_\infty$-coefficients we prove linear convergence in the $H^1$-norm, also for materials with large Lam\'{e} parameter $\lambda$. The theoretical a priori error estimate is confirmed by numerical examples.

math.NA↗

arXiv2015

Multiscale techniques for parabolic equations

We use the local orthogonal decomposition technique to derive a generalized finite element method for linear and semilinear parabolic equations with spatial multiscale diffusion coefficient. 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 in the diffusion coefficient, is proven in the $L_\infty(L_2)$-norm. We present numerical examples, which confirm our theoretical findings.

math.NA↗