arXiv subjects

Van Chien Le

Publications and source records attributed to Van Chien Le.

11 recordsLinked to original sources

arXiv2026

Quasi-Helmholtz Calderón Multiplicative Preconditioning for Higher-Order Global Multi-Trace Integral Equations

The paper presents a higher-order global multi-trace integral equation for time-harmonic electromagnetic scattering by composite objects. The higher-order multi-trace formulation is preconditioned with a Calderón multiplicative preconditioner using higher-order quasi-Helmholtz projectors. The higher-order quasi-Helmholtz projectors separate the solenoidal and non-solenoidal components of the basis functions. Separate access to the Helmholtz components circumvents the explicit inversion of an ill-conditioned mixed Gram matrix. This enables the application of Calderón multiplicative preconditioning without refining the mesh and resorting to dual basis functions. Furthermore, it enables low-frequency stabilization, as the solenoidal and non-solenoidal components can be rescaled individually. The higher-order quasi-Helmholtz projectors are computed iteratively, enabling iterative solvers to efficiently solve the preconditioned matrix system, yielding accurate solutions in a few dozen iterations. Numerical experiments are conducted for composite dielectric bodies, confirming the effectiveness of the proposed preconditioner for the global multi-trace formulation discretized with higher-order basis functions for dense meshes and at very low frequencies

cs.CE↗

arXiv2026

A Stable, Accurate, and Well-Conditioned Time-Domain PMCHWT Formulation

This paper introduces a new boundary element formulation for transient electromagnetic scattering by homogeneous dielectric objects, based on the time-domain PMCHWT equation. To address dense-mesh breakdown, a multiplicative Calderón preconditioner constructed from a modified static electric field integral operator is employed. Large-timestep breakdown and late-time instability are simultaneously resolved through a rescaling of the Helmholtz components using quasi-Helmholtz projectors, with temporal differentiation and integration serving as the rescaling operators. This rescaling additionally balances the loop and star components in the large-timestep regime, thereby preventing loss of accuracy in the secondary quantities caused by numerical cancellation. The resulting discrete system is solved using a marching-on-in-time scheme in conjunction with iterative solvers. Numerical experiments for simply- and multiply-connected dielectric scatterers, including highly non-smooth geometries, corroborate the stability and efficiency of the proposed approach and demonstrate its ability to produce accurate derived quantities in the large-timestep regime.

eess.SY↗

arXiv2026

Multitrace Müller Boundary Integral Equation for Electromagnetic Scattering by Composite Objects

This paper introduces a boundary integral equation for time-harmonic electromagnetic scattering by composite dielectric objects. The formulation extends the classical Müller equation to composite structures through the global multitrace method. The key ingredient enabling this extension is the use of the Stratton-Chu representation in complementary region, also known as the extinction property, which augments the off-diagonal blocks of the interior representation operator. The resulting block system is composed entirely of second-kind operators. A Petrov-Galerkin (mixed) discretization using Rao-Wilton-Glisson trial functions and Buffa-Christiansen test functions is employed, yielding linear systems that remain well conditioned on dense meshes and at low frequencies without the need for additional stabilization. This reduces computational costs associated with matrix-vector multiplications and iterative solving. Numerical experiments demonstrate the accuracy of the method in computing field traces and derived quantities.

math.NA↗

arXiv2025

Parabolic PDEs on a fixed domain with evolving subdomains: function spaces and well-posedness

This paper develops the necessary ingredients for the variational approach of initial boundary-value problems of parabolic partial differential equations on a fixed spatial domain containing evolving subdomains. In particular, we introduce function spaces for the variational solution that extend standard Sobolev-Bochner spaces to account for a coefficient associated with the time derivative that may be discontinuous across the evolving interface. We further show the density of smooth functions in these spaces by extending the mollification technique and the Reynolds transport theorem, and establish the corresponding "embedding" theory and an integration by parts formula. Finally, we prove the well-posedness of the space-time variational formulation in the natural setting using the Banach-Necas-Babuska theorem.

math.AP↗

arXiv2025

On the Late-Time Instability of MOT solution to the Time-Domain PMCHWT Equation

This paper investigates the late-time instability of marching-on-in-time solution to the time-domain PMCHWT equation. The stability analysis identifies the static solenoidal nullspace of the time-domain electric field integral operator as the primary cause of instability. Furthermore, it reveals that the instability mechanisms of the time-domain PMCHWT equation are fundamentally different from those of the time-domain electric field integral equation. In particular, the PMCHWT's instability is much more sensitive to numerical quadrature errors, and its spectral characteristics are strongly influenced by the topology and smoothness of the scatterer surface.

math.NA↗

arXiv2025

A space-time interface-fitted method for moving-subdomain distributed control problems with energy regularization

This paper investigates a space-time interface-fitted approximation of a moving-interface optimal control problem with energy regularization. We reformulate the optimality conditions into a variational problem involving both the state and adjoint. This problem is shown to be equivalent to our optimal control problem. Based on fully unstructured, space-time interface-fitted meshes, we propose and analyze a Petrov-Galerkin approximation of the problem. An optimal error estimate with respect to a discrete norm is established under a specific regularity assumption on the state and adjoint. Several numerical results are presented to corroborate our theoretical results.

math.OC↗

arXiv2025

A fitted space-time finite element method for an advection-diffusion problem with moving interfaces

This paper presents a space-time interface-fitted finite element method for solving a parabolic advection-diffusion problem with a nonstationary interface. The jumping diffusion coefficient gives rise to the discontinuity of the solution gradient across the interface. We use the Banach-Necas-Babuska theorem to show the well-posedness of the continuous variational problem. A fully discrete finite-element based scheme is analyzed using the Galerkin method and unstructured interface-fitted meshes. An optimal error estimate is established in a discrete energy norm under a globally low but locally high regularity condition. Some numerical results corroborate our theoretical results.

math.NA↗

arXiv2024

An operator preconditioned combined field integral equation for electromagnetic scattering

This paper aims to address two issues of integral equations for the scattering of time-harmonic electromagnetic waves by a perfect electric conductor with Lipschitz continuous boundary: ill-conditioned {boundary element Galerkin matrices} on fine meshes and instability at spurious resonant frequencies. The remedy to ill-conditioned matrices is operator preconditioning, and resonant instability is eliminated by means of a combined field integral equation. Exterior traces of single and double layer potentials are complemented by their interior counterparts for a purely imaginary wave number. We derive the corresponding variational formulation in the natural trace space for electromagnetic fields and establish its well-posedness for all wave numbers. A Galerkin discretization scheme is employed using conforming edge boundary elements on dual meshes, which produces well-conditioned discrete linear systems of the variational formulation. Some numerical results are also provided to support the numerical analysis.

math.NA↗

arXiv2024

Boundary element methods for the magnetic field integral equation on polyhedra

This paper provides a rigorous analysis of boundary element methods for the magnetic field integral equation on Lipschitz polyhedra. The magnetic field integral equation is widely used in practical applications to model electromagnetic scattering by a perfectly conducting body. The governing operator is shown to be coercive by means of the electric field integral operator with a purely imaginary wave number. Consequently, the continuous variational problem is uniquely solvable, given that the wave number does not belong to the spectrum of the interior Maxwell's problem. A Petrov-Galerkin discretization scheme is then introduced, employing Raviart-Thomas boundary elements for the solution space and Buffa-Christiansen boundary elements for the test space. Under a mild assumption depending only on the geometrical domain, the corresponding discrete inf-sup condition is proven, implying the unique solvability of the discrete problem. An asymptotically quasi-optimal error estimate for numerical solutions is established, and the convergence rate of the numerical scheme is examined. In addition, the resulting matrix system is shown to be well-conditioned regardless of the mesh refinement. Some numerical results are presented to support the theoretical analysis.

math.NA↗

arXiv2024

A stabilized time-domain combined field integral equation using the quasi-Helmholtz projectors

This paper introduces a time-domain combined field integral equation for electromagnetic scattering by a perfect electric conductor. The new equation is obtained by leveraging the quasi-Helmholtz projectors, which separate both the unknown and the source fields into solenoidal and irrotational components. These two components are then appropriately rescaled to cure the solution from a loss of accuracy occurring when the time step is large. Yukawa-type integral operators of a purely imaginary wave number are also used as a Calderon preconditioner to eliminate the ill-conditioning of matrix systems. The stabilized time-domain electric and magnetic field integral equations are linearly combined in a Calderon-like fashion, then temporally discretized using an appropriate pair of trial functions, resulting in a marching-on-in-time linear system. The novel formulation is immune to spurious resonances, dense discretization breakdown, large-time step breakdown and dc instabilities stemming from non-trivial kernels. Numerical results for both simply-connected and multiply-connected scatterers corroborate the theoretical analysis.

math.NA↗

arXiv2024

A numerical scheme for solving an induction heating problem with moving non-magnetic conductor

This paper investigates an induction heating problem in a multi-component system containing a moving non-magnetic conductor. The electromagnetic process is described by the eddy current model, and the heat transfer process is governed by the convection-diffusion equation. Both processes are coupled by a restrained Joule heat source. A temporal discretization scheme is introduced to solve the corresponding variational system numerically. With the aid of the Reynolds transport theorem, we prove the convergence of the proposed scheme as well as the well-posedness of the variational problem. Some numerical experiments are also performed to assess the performance of the numerical scheme.

math.NA↗