arXiv subjects

Ignacio Labarca-Figueroa

Publications and source records attributed to Ignacio Labarca-Figueroa.

6 recordsLinked to original sources

arXiv2026

Numerical approach to the London Equation of superconductivity

In this work, we propose a general discretization strategy for solving the London equation for type-II superconductors in the whole space $\mathbb{R}^3$. To compute the magnetic field $H_0$, we reformulate the problem for the magnetic potential as a transmission problem and discretize it through a nonstandard FEM-BEM coupling. This formulation accounts for both the bounded interior domain and the unbounded exterior domain without introducing an artificial truncation. We then compute the vector field $B_0$, which arises from the Helmholtz-Hodge decomposition of the magnetic potential in the superconducting sample. This field enters the isoflux problem, which identifies the curves along which vortex nucleation first becomes energetically favorable in the Ginzburg--Landau model of superconductivity. We recast the equations for $B_0$ using the mixed formulation of Kikuchi, in which the divergence-free constraint is imposed weakly, and discretize the resulting problem using a classical $H(\operatorname{curl})$-conforming finite element discretization. We validate our discretization strategy through convergence tests and conclude with an application to the isoflux problem. For a ball under a constant applied magnetic field, the unique maximizer is the diameter aligned with the field. For ellipsoids under a constant applied magnetic field aligned with their major axis, our computations provide numerical evidence of a different behavior in sufficiently elongated, cigar-shaped geometries: off-axis competitors reminiscent of U-shaped vortex configurations attain a larger isoflux ratio than the major axis. Since the major axis is therefore not a maximizer, any off-axis maximizer generates, by rotational symmetry, a continuous family of equivalent configurations, implying non-uniqueness and the presence of a degenerate rotational direction in the isoflux problem.

math.NA↗

arXiv2026

Robust Hierarchical Matrix Compression of Acoustic Volume and Boundary Integral Operators

Discretizing integral formulations of the Helmholtz equation yields dense linear systems. Hence, simulating acoustic models at larger scales or higher frequencies is typically constrained by memory capacity. Fast algorithms, such as hierarchical matrix compression, reduce the memory footprint substantially while controlling the approximation error in matrix-vector multiplications. However, the commonly used Adaptive Cross Approximation suffers from early-convergence problems, where the iterative construction of low-rank decompositions stops before reaching the targeted error tolerance. This failure arises when the error estimator does not capture significant components of the matrix structure under partial pivoting. This manuscript proposes a new diagonal convergence criterion, additional matrix elements for the pivoting strategy, an extended admissibility condition, and a sustained convergence check to improve the robustness of hierarchical matrix compression. These modifications improve compression reliability without increasing memory. We tested our compression strategy on various discretized volume and boundary integral operators. The computational results show that our approach successfully compresses all benchmark matrices within predefined tolerances, thereby resolving the early-convergence issues encountered in standard algorithms. This robust matrix compression was achieved at the same memory footprint as alternative compression strategies. Furthermore, a complexity analysis shows log-linear memory scaling with mesh refinement at constant frequency. Finally, we successfully applied our robust matrix compression algorithm to a coupled system of volume and boundary integral operators that models transcranial ultrasound propagation. This confirms the feasibility of our robust algorithm to accelerate large-scale simulations with high-resolution meshes in a biomedical application.

math.NA↗

arXiv2026

Efficient boundary elements for the Smoluchowski diffusion equation

The Smoluchowski diffusion equation describes diffusion in the presence of external forces. Studying the mechanical response of soft materials to linear forces, such as shear, results in a boundary value problem involving an Ornstein-Uhlenbeck operator in an exterior domain with non-constant, unbounded coefficients. In this article, we present efficient and highly accurate boundary element methods in the frequency domain, motivated by applications in soft matter physics. Our key contributions concern the accurate assembly of the Galerkin matrix, combining the approximation of the fundamental solution as a Fourier integral with the resolution of near-field singularities. Numerical experiments demonstrate the accuracy and efficiency of the proposed methods and show their relevance for the computation of rheological quantities.

math.NA↗

arXiv2025

A posteriori error estimates and space-adaptive mesh refinements for time-dependent scattering problems

This work studies a posteriori error estimates and their use for time-dependent acoustic scattering problems, formulated as a time-dependent boundary integral equation based on a single-layer ansatz. The integral equation is discretized by the convolution quadrature method in time and by boundary elements in space. We prove the reliability of an error estimator of residual type and study the resulting space-adaptive mesh refinements. Moreover, we present a simple modification of the convolution quadrature method based on temporal shifts, which recovers, for the boundary densities, the full classical temporal convergence order $2m-1$ of the temporal convolution quadrature method based on the $m$-stage convolution quadrature semi-discretization. We numerically observe that the adaptive scheme yields asymptotically optimal meshes for an acoustic scattering problem in two dimensions.

math.NA↗

arXiv2024

Domain Uncertainty Quantification for the Lippmann-Schwinger Volume Integral Equation

In this work, we consider the propagation of acoustic waves in unbounded domains characterized by a constant wavenumber, except possibly in a bounded region. The geometry of this inhomogeneity is assumed to be uncertain, and we are particularly interested in studying the propagation of this behavior throughout the physical model considered. A key step in our analysis consists of recasting the physical model-originally set in an unbounded domain-into a computationally manageable formulation based on Volume Integral Equations (VIEs), particularly the Lippmann-Schwinger equation. We show that both the leading operator in this volume integral formulation and its solution depend holomorphically on shape variations of the support of the aforementioned inhomogeneity. This property, known as shape holomorphy, is crucial in the analysis and implementation of various methods used in computational Uncertainty Quantification (UQ). We explore the implications of this result in forward and inverse UQ and provide numerical experiments illustrating and confirming the theoretical predictions.

math.NA↗

arXiv2024

Coupled Boundary and Volume Integral Equations for Electromagnetic Scattering

We study frequency domain electromagnetic scattering at a bounded, penetrable, and inhomogeneous obstacle $ \Omega \subset \mathbb{R}^3 $. From the Stratton-Chu integral representation, we derive a new representation formula when constant reference coefficients are given for the interior domain. The resulting integral representation contains the usual layer potentials, but also volume potentials on $\Omega$. Then it is possible to follow a single-trace approach to obtain boundary integral equations perturbed by traces of compact volume integral operators with weakly singular kernels. The coupled boundary and volume integral equations are discretized with a Galerkin approach with usual Curl-conforming and Div-conforming finite elements on the boundary and in the volume. Compression techniques and special quadrature rules for singular integrands are required for an efficient and accurate method. Numerical experiments provide evidence that our new formulation enjoys promising properties.

math.NA↗