arXiv subjects

Liet Vo

Publications and source records attributed to Liet Vo.

18 recordsLinked to original sources

arXiv2026

A splitting mixed finite element method for a stochastic Keller-Segel system with multiplicative noise

In this paper, we propose and analyze a splitting mixed finite element method for a stochastic Keller--Segel system with logistic growth driven by multiplicative noise. By introducing an auxiliary variable representing the chemical gradient together with a time-lagged splitting strategy, the proposed method decouples the original coupled system into a sequence of simpler subproblems. Consequently, it eliminates the Ladyzhenskaya--Babu\v{s}ka--Brezzi stability constraint, permits the use of continuous piecewise linear finite element spaces for all unknowns, and avoids solving a fully coupled nonlinear system at each time step, thereby significantly reducing the computational cost. Combined with an implicit Euler time discretization, the proposed approach yields a fully discrete numerical scheme for the stochastic Keller--Segel system. Using a localization technique together with suitable stochastic stability arguments, we establish optimal strong error estimates for the fully discrete approximations and prove convergence in probability with explicit convergence rates. Numerical experiments verify the theoretical convergence rates and demonstrate that the proposed method successfully captures the global boundedness induced by the logistic growth term as well as the influence of multiplicative noise on chemotactic aggregation.

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arXiv2026

A High-Order Perturbation of Envelopes (HOPE) Method for Electromagnetic Scattering by Periodic Inhomogeneous Chiral Media

Chirality plays an important role in many optical phenomena, and it is crucial to have efficient and accurate numerical algorithms to simulate solutions in this setting. In this paper, we discuss a High-Order Spectral method coupled to geometric regular perturbation theory, which results in just such a method. We view the chirality of the laterally periodic, constant permittivity/permeability structure as deviating from a background value, and demonstrate analyticity of the field scattered by a plane electromagnetic wave with respect to this deformation. This analyticity is not only with respect to chirality deformations of arbitrarily large real size, but also joint in the spatial variables. We also show how the High--Order Perturbation of Envelopes (HOPE) recursions, which we used to establish these results, can be implemented as a rapid, robust, and reliable numerical algorithm.

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arXiv2026

Finite element approximations of the stochastic Benjamin-Bona-Mahony equation with multiplicative noise

This paper is devoted to the numerical analysis of a fully discrete finite element approximation for the stochastic Benjamin-Bona-Mahony equation driven by multiplicative noise. We first establish the existence and uniqueness of solutions to the stochastic BBM equation within an appropriate variational framework and derive several stability estimates for the continuous problem, including an exponential stability result. For the numerical approximation, a conforming finite element method is employed for spatial discretization and is coupled with the implicit Euler-Maruyama scheme for time integration. The convergence of the fully discrete scheme is investigated under two different classes of multiplicative noise. When the noise coefficient is bounded, we obtain optimal strong error estimates in full expectation by combining exponential stability properties of both the stochastic BBM solution and its fully discrete counterpart with a stochastic Gronwall inequality. In the case of general multiplicative noise, where boundedness assumptions are no longer valid, a localization technique based on high-probability events in the sample space is introduced, leading to sub-optimal convergence rates in probability. Finally, numerical experiments are presented to corroborate the theoretical results and to demonstrate the performance of the proposed method.

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arXiv2025

A mixed finite element method for the stochastic Boussinesq equations with multiplicative noise

This work investigates a fully discrete mixed finite element method for the stochastic Boussinesq system driven by multiplicative noise. The spatial discretization is performed using a standard mixed finite element method, while the temporal discretization is based on a semi-implicit Euler-Maruyama scheme. By combining a localization technique with high-moment stability estimates, we establish error bounds for the velocity, pressure, and temperature approximations. As a direct consequence, we prove convergence in probability for the fully discrete method in both $L^2$ and $H^1$-type norms. Several numerical experiments are presented to validate the theoretical error estimates and demonstrate the effectiveness of the proposed scheme.

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arXiv2025

Full moment error estimates in strong norms for numerical approximations of stochastic Navier-Stokes equations with multiplicative noise, Part I: time discretization

This paper focuses on deriving optimal-order full moment error estimates in strong norms for both velocity and pressure approximations in the Euler-Maruyama time discretization of the stochastic Navier-Stokes equations with multiplicative noise. Additionally, it introduces a novel approach and framework for the numerical analysis of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noise in general. The main ideas of this approach include establishing exponential stability estimates for the SPDE solution, leveraging a discrete stochastic Gronwall inequality, and employing a bootstrap argument.

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arXiv2025

Fully discrete finite element methods for the stochastic Kuramoto-Sivashinsky equation with multiplicative noise

We investigate a fully discrete finite element approximation for the stochastic Kuramoto-Sivashinsky equation, combining the standard finite element methods in spatial discretization with the implicit Euler-Maruyama scheme in time. Rigorous error estimates are established for two distinct noise regimes. In the case of bounded multiplicative noise, we prove optimal strong convergence rates in full expectation. The analysis relies crucially on a stochastic Gronwall inequality and an exponential stability estimate for the PDE solution, which together control the interplay between the nonlinear drift and the multiplicative stochastic forcing. For general multiplicative noise, where boundedness no longer holds, we derive sub-optimal convergence rates in probability by introducing a localization technique based on carefully constructed subsets of the sample space. This dual framework demonstrates that the proposed fully discrete scheme achieves strong convergence under bounded noise and probabilistic convergence under general multiplicative noise, thus providing the first comprehensive error analysis for numerical approximations of the stochastic Kuramoto-Sivashinsky equation.

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arXiv2025

Analysis of fully discrete Crank-Nicolson finite element methods for a stochastic Keller-Segel chemotaxis system with gradient-type multiplicative noise

We develop and analyze numerical methods for a stochastic Keller-Segel system perturbed by Stratonovich noise, which models chemotactic behavior under randomly fluctuating environmental conditions. The proposed fully discrete scheme couples a Crank-Nicolson time discretization with a splitting mixed finite element method in space. We rigorously prove the stability of the numerical scheme and establish strong convergence rates of order $O(k^{1/2} + k^{-1/2}h^2)$, where $k$ and $h$ denote the time and spatial step sizes, respectively. Notably, the presence of stochastic forcing leads to an inverse dependence on $k$ in the error estimates, distinguishing the convergence behavior from that of the deterministic case. Numerical experiments are presented to validate the theoretical results and demonstrate the effectiveness and accuracy of the proposed methods.

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arXiv2025

Optimal Order Space-Time Discretization Methods for the Nonlinear Stochastic Elastic Wave Equations with Multiplicative Noise

This paper develops and analyzes an optimal-order semi-discrete scheme and its fully discrete finite element approximation for nonlinear stochastic elastic wave equations with multiplicative noise. A non-standard time-stepping scheme is introduced for time discretization, it is showed that the scheme converges with rates $O(\tau)$ and $O(\tau^{\frac32})$ respectively in the energy- and $L^2$-norm, which are optimal with respect to the time regularity of the PDE solution. For spatial discretization, the standard finite element method is employed. It is proven that the fully discrete method converges with optimal rates $O(\tau + h)$ and $O(\tau^{\frac{3}{2}} + h^2)$ respectively in the energy- and $L^2$-norm. The cruxes of the analysis are to establish some high-moment stability results and utilize a refined error estimate for the trapezoidal quadrature rule to control the nonlinearities from the drift term and the multiplicative noise. Numerical experiments are also provided to validate the theoretical results.

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arXiv2024

Optimal order time discretizations for stochastic semilinear wave equations with multiplicative noise

This paper is concerned with developing and analyzing two novel implicit temporal discretization methods for the stochastic semilinear wave equations with multiplicative noise. The proposed methods are natural extensions of well-known time-discrete schemes for deterministic wave equations, hence, they are easy to implement. It is proved that both methods are energy-stable. Moreover, the first method is shown to converge with the linear order in the energy norm, while the second method converges with the $\mathcal{O}(\tau^{\frac32})$ order in the $L^2$-norm, which is optimal with respect to the time regularity of the solution to the underlying stochastic PDE. The convergence analyses of both methods, which are different and quite involved, require some novel numerical techniques to overcome difficulties caused by the nonlinear noise term and the interplay between nonlinear drift and diffusion. Numerical experiments are provided to validate the sharpness of the theoretical error estimate results.

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arXiv2024

A High-Order Perturbation of Envelopes (HOPE) Method for Vector Electromagnetic Scattering by Periodic Inhomogeneous Media: Analytic Continuation

Electromagnetic waves interacting with three--dimensional periodic structures occur in many applications of great scientific and engineering interest. These three dimensional interactions are extremely complicated and subtle, so it is unsurprising that practitioners find their rapid, robust, and accurate numerical simulation to be of paramount interest. Among the wide array of possible numerical approaches, the High--Order Spectral algorithms are often preferred due to their surpassing fidelity with a moderate number of unknowns, and here we describe an algorithm that fits into this class. In addition, we take a perturbative approach to the problem which views the deviation of the permittivity from a reference value as the deformation and we conduct a regular perturbation theory. This work concludes a line of research on these methods which began with two-dimensional problems governed by the Helmholtz equation and moved to small perturbations in the fully three-dimensional vector Maxwell equations. We now extend these latter results to large (real) perturbations constituting a rigorous analytic continuation.

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arXiv2023

A High-Order Perturbation of Envelopes (HOPE) Method for Vector Electromagnetic Scattering by Periodic Inhomogeneous Media

The scattering of electromagnetic waves by three--dimensional periodic structures is important for many problems of crucial scientific and engineering interest. Due to the complexity and three-dimensional nature of these waves, the fast, accurate, and reliable numerical simulations of these are indispensable for engineers and scientists alike. For this, High Order Spectral methods are frequently employed and here we describe an algorithm in this class. Our approach is perturbative in nature where we view the deviation of the permittivity from a constant value as the deformation and we pursue regular perturbation theory. This work extends our previous contribution regarding the Helmholtz equation to the full vector Maxwell equations, by providing a rigorous analyticity theory, both in deformation size and spatial variable (provided that the permittivity is, itself, analytic).

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arXiv2023

Higher order time discretization method for a class of semilinear stochastic partial differential equations with multiplicative noise

In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the diffusion term is assumed to be globally Lipschitz continuous. Our new strategy for time discretization is based on the Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of nearly $1$ for the approximate solution. The proof is based on new H\"older continuity estimates of the SPDE solution and the nonlinear term. For the general polynomial-type drift term, there are difficulties in deriving even the stability of the numerical solutions. We propose an interpolation-based finite element method for spatial discretization to overcome the difficulties. Then we obtain $H^1$ stability, higher moment $H^1$ stability, $L^2$ stability, and higher moment $L^2$ stability results using numerical and stochastic techniques. The nearly optimal convergence orders in time and space are hence obtained by coupling all previous results. Numerical experiments are presented to implement the proposed numerical scheme and to validate the theoretical results.

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arXiv2022

Higher order time discretization method for the stochastic Stokes equations with multiplicative noise

In this paper, we propose a new approach for the time-discretization of the incompressible stochastic Stokes equations with multiplicative noise. Our new strategy is based on the classical Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of at most $1$ for both velocity and pressure approximations. The proof is based on a new Hölder continuity estimate of the velocity solution. While the errors of the velocity approximation are estimated in the standard $L^2$- and $H^1$-norms, the pressure errors are carefully analyzed in a special norm because of the low regularity of the pressure solution. In addition, a new interpretation of the pressure solution, which is very useful in computation, is also introduced. Numerical experiments are also provided to validate the error estimates and their sharpness.

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arXiv2022

High moment and pathwise error estimates for fully discrete mixed finite element approximattions of stochastic Navier-Stokes equations with additive noise

This paper is concerned with high moment and pathwise error estimates for fully discrete mixed finite element approximattions of stochastic Navier-Stokes equations with general additive noise. The implicit Euler-Maruyama scheme and standard mixed finite element methods are employed respectively for the time and space discretizations. High moment error estimates for both velocity and a time-avraged pressure approximations in strong $L^2$ and energy norms are obtained, pathwise error estimates are derived by using the Kolmogorov Theorem. Unlike their derterministic counterparts, the spatial error constants grow in the order of $O(k^{-\frac12})$, where $k$ denotes time step size. Numerical experiments are also provided to validate the error estimates and their sharpness.

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arXiv2021

An efficient iterative method for solving parameter-dependent and random convention-diffusion problems

This paper develops and analyzes a general iterative framework for solving parameter-dependent and random convection-diffusion problems. It is inspired by the multi-modes method of [7,8] and the ensemble method of [20] and extends those methods into a more general and unified framework. The main idea of the framework is to reformulate the underlying problem into another problem with parameter-independent convection and diffusion coefficients and a parameter-dependent (and solution-dependent) right-hand side, a fixed-point iteration is then employed to compute the solution of the reformulated problem. The main benefit of the proposed approach is that an efficient direct solver and a block Krylov subspace iterative solver can be used at each iteration, allowing to reuse the $LU$ matrix factorization or to do an efficient matrix-matrix multiplication for all parameters, which in turn results in significant computation saving. Convergence and rates of convergence are established for the iterative method both at the variational continuous level and at the finite element discrete level under some structure conditions. Several strategies for establishing reformulations of parameter-dependent and random diffusion and convection-diffusion problems are proposed and their computational complexity is analyzed. Several 1-D and 2-D numerical experiments are also provided to demonstrate the efficiency of the proposed iterative method and to validate the theoretical convergence results.

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arXiv2021

Analysis of Chorin-Type Projection Methods for the Stochastic Stokes Equations with General Multiplicative Noises

This paper is concerned with numerical analysis of two fully discrete Chorin-type projection methods for the stochastic Stokes equations with general non-solenoidal multiplicative noise. The first scheme is the standard Chorin scheme and the second one is a modified Chorin scheme which is designed by employing the Helmholtz decomposition on the noise function at each time step to produce a projected divergence-free noise and a "pseudo pressure" after combining the original pressure and the curl-free part of the decomposition. Optimal order rates of the convergence are proved for both velocity and pressure approximations of these two (semi-discrete) Chorin schemes. It is crucial to measure the errors in appropriate norms. The fully discrete finite element methods are formulated by discretizing both semi-discrete Chorin schemes in space by the standard finite element method. Suboptimal order error estimates are derived for both fully discrete methods. It is proved that all spatial error constants contain a growth factor $k^{-1/2}$, where $k$ denotes the time step size, which explains the deteriorating performance of the standard Chorin scheme when $k\to 0$ and the space mesh size is fixed as observed earlier in the numerical tests of [9]. Numerical results are also provided to guage the performance of the proposed numerical methods and to validate the sharpness of the theoretical error estimates.

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arXiv2021

High moment and pathwise error estimates for fully discrete mixed finite element approximations of the Stochastic Stokes Equations with Multiplicative Noises

This paper is concerned with high moment and pathwise error estimates for both velocity and pressure approximations of the Euler-Maruyama scheme for time discretization and its two fully discrete mixed finite element discretizations. The main idea for deriving the high moment error estimates for the velocity approximation is to use a bootstrap technique starting from the second moment error estimate. The pathwise error estimate, which is sub-optimal in the energy norm, is obtained by using Kolmogorov's theorem based on the high moment error estimates. Unlike for the velocity error estimate, the higher moment and pathwise error estimates for the pressure approximation are derived in a time-averaged norm. In addition, the impact of noise types on the rates of convergence for both velocity and pressure approximations is also addressed.

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arXiv2020

Optimally Convergent Mixed Finite Element Methods for the Stochastic Stokes Equations

We propose some new mixed finite element methods for the time dependent stochastic Stokes equations with multiplicative noise, which use the Helmholtz decomposition of the driving multiplicative noise. It is known [16] that the pressure solution has a low regularity, which manifests in sub-optimal convergence rates for well-known inf-sup stable mixed finite element methods in numerical simulations, see [10]. We show that eliminating this gradient part from the noise in the numerical scheme leads to optimally convergent mixed finite element methods, and that this conceptual idea may be used to retool numerical methods that are well-known in the deterministic setting, including pressure stabilization methods, so that their optimal convergence properties can still be maintained in the stochastic setting. Computational experiments are also provided to validate the theoretical results and to illustrate the conceptional usefulness of the proposed numerical approach.

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