SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2172-2195, October 2024. Abstract. A novel evolving surface finite element method, based on a novel equivalent formulation of the continuous problem, is proposed for computing the evolution of a closed hypersurface moving under a prescribed velocity field in two- and three-dimensional spaces. The method improves the mesh quality of the approximate

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2143-2171, October 2024. Abstract. The coupling of nonconservative hyperbolic systems at a static interface has been a delicate issue as common approaches rely on the Lax-curves of the systems, which are not always available. To address this a new linear relaxation system is introduced, in which a nonlocal source term accounts for the nonconservative

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2121-2142, October 2024. Abstract. This paper studies the discretization of a homogenization and dimension reduction model for the elastic deformation of microstructured thin plates proposed by Hornung, Neukamm, and Velčić [Calc. Var. Partial Differential Equations, 51 (2014), pp. 677–699]. Thereby, a nonlinear bending energy is based on

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2087-2120, October 2024. Abstract. Along with the practical success of the discovery of dynamics using deep learning, the theoretical analysis of this approach has attracted increasing attention. Prior works have established the grid error estimation with auxiliary conditions for the discovery of dynamics using linear multistep methods and

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2071-2086, October 2024. Abstract. For the numerical solution of the cubic nonlinear Schrödinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 2048-2070, August 2024. Abstract. We present new Dirichlet–Neumann and Neumann–Dirichlet algorithms with a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semidiscretization, we use the Lagrange multiplier approach to derive a coupled forward-backward optimality system, which can then

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 2025-2047, August 2024. Abstract. Statistical inverse learning aims at recovering an unknown function [math] from randomly scattered and possibly noisy point evaluations of another function [math], connected to [math] via an ill-posed mathematical model. In this paper we blend statistical inverse learning theory with the classical regularization

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 2004-2024, August 2024. Abstract. We propose and analyze a novel approach to construct structure preserving approximations for the Poisson–Nernst–Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1979-2003, August 2024. Abstract. We introduce a new overlapping domain decomposition method (DDM) to solve fully nonlinear elliptic partial differential equations (PDEs) approximated with monotone schemes. While DDMs have been extensively studied for linear problems, their application to fully nonlinear PDEs remains limited in the literature

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1956-1978, August 2024. Abstract. In this article, we employ the construction of the time-marching discontinuous Petrov–Galerkin (DPG) scheme we developed for linear problems to derive high-order multistage DPG methods for nonlinear systems of ordinary differential equations. The methodology extends to abstract evolution equations in Banach

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1929-1955, August 2024. Abstract. In this work is considered an elliptic problem, referred to as the Ventcel problem, involving a second-order term on the domain boundary (the Laplace–Beltrami operator). A variational formulation of the Ventcel problem is studied, leading to a finite element discretization. The focus is on the construction

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1901-1928, August 2024. Abstract. We propose and analyze a novel symmetric Gautschi-type exponential wave integrator (sEWI) for the nonlinear Schrödinger equation (NLSE) with low regularity potential and typical power-type nonlinearity of the form [math] with [math] being the wave function and [math] being the exponent of the nonlinearity

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1874-1900, August 2024. Abstract. We investigate a class of parametric elliptic eigenvalue problems with homogeneous essential boundary conditions, where the coefficients (and hence the solution) may depend on a parameter. For the efficient approximate evaluation of parameter sensitivities of the first eigenpairs on the entire parameter space

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1844-1873, August 2024. Abstract. Multiscale partial differential equations (PDEs) arise in various applications, and several schemes have been developed to solve them efficiently. Homogenization theory is a powerful methodology that eliminates the small-scale dependence, resulting in simplified equations that are computationally tractable

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1814-1843, August 2024. Abstract. We propose and analyze discontinuous Galerkin (dG) approximations to 3D−1D coupled systems which model diffusion in a 3D domain containing a small inclusion reduced to its 1D centerline. Convergence to weak solutions of a steady state problem is established via deriving a posteriori error estimates and bounds

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1782-1813, August 2024. Abstract. Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure displacement have been extensively studied for incompressible fluid-structure interactions (FSIs) over the past decade. While these methods are known to be stable and easy to implement, optimal error analysis

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1759-1781, August 2024. Abstract. Motivated by polynomial approximations of differential forms, we study analytical and numerical properties of a polynomial interpolation problem that relies on function averages over interval segments. The usage of segment data gives rise to new theoretical and practical aspects that distinguish this problem

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1736-1758, August 2024. Abstract. We consider nonlinear delay differential and renewal equations with infinite delay. We extend the work of Gyllenberg et al. [Appl. Math. Comput., 333 (2018), pp. 490–505] by introducing a unifying abstract framework, and we derive a finite-dimensional approximating system via pseudospectral discretization

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1713-1735, August 2024. Abstract. Quasiperiodic systems, related to irrational numbers, are space-filling structures without decay or translation invariance. How to accurately recover these systems, especially for low-regularity cases, presents a big challenge in numerical computation. In this paper, we propose a new algorithm, the finite

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1687-1712, August 2024. Abstract. In this paper we study the a posteriori bounds for a conforming piecewise linear finite element approximation of the Signorini problem. We prove new rigorous a posteriori estimates of residual type in [math], for [math] in two spatial dimensions. This new analysis treats the positive and negative parts of

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1660-1686, August 2024. Abstract. We propose a finite element discretization for the steady, generalized Navier–Stokes equations for fluids with shear-dependent viscosity, completed with inhomogeneous Dirichlet boundary conditions and an inhomogeneous divergence constraint. We establish (weak) convergence of discrete solutions as well as

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1638-1659, August 2024. Abstract. Maximal regularity is a kind of a priori estimate for parabolic-type equations, and it plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) time-stepping

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1609-1637, August 2024. Abstract. When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higher-order schemes, making it impractical to use high-order schemes for stiff problems. We construct in this paper a new class of BDF and implicit-explicit

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1589-1608, August 2024. Abstract. This work proposes a discretization of the acoustic wave equation with possibly oscillatory coefficients based on a superposition of discrete solutions to spatially localized subproblems computed with an implicit time discretization. Based on exponentially decaying entries of the global system matrices and

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1565-1588, August 2024. Abstract. The dispersion error is currently the dominant error for computed solutions of wave propagation problems with high-frequency components. In this paper, we define and give explicit examples of interior [math]-dispersion-relation-preserving schemes, of interior order of accuracy 4, 5, 6, and 7, with a complete

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1539-1564, August 2024. Abstract. We introduce a pressure robust finite element method for the linearized magnetohydrodynamics equations in three space dimensions, which is provably quasi-robust also in the presence of high fluid and magnetic Reynolds numbers. The proposed scheme uses a nonconforming BDM approach with suitable DG terms for

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1515-1538, August 2024. Abstract. In approximation of functions based on point values, least-squares methods provide more stability than interpolation, at the expense of increasing the sampling budget. We show that near-optimal approximation error can nevertheless be achieved, in an expected [math] sense, as soon as the sample size [math]

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1492-1514, August 2024. Abstract. This paper derives a discrete dual problem for a prototypical hybrid high-order method for convex minimization problems. The discrete primal and dual problem satisfy a weak convex duality that leads to a priori error estimates with convergence rates under additional smoothness assumptions. This duality holds

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1465-1491, August 2024. Abstract. A robust substructuring type preconditioner is developed for high order approximation of problem for which the element matrix takes the form [math] where [math] and [math] are the mass and stiffness matrices, respectively. A standard preconditioner for the pure stiffness matrix results in a condition number

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1443-1464, June 2024. Abstract. In this work we connect machine learning techniques, in particular kernel machine learning, to inverse source and scattering problems. We show the proposed kernel machine learning has demonstrated generalization capability and has a rigorous mathematical foundation. The proposed learning is based on the Mercer

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1420-1442, June 2024. Abstract. In this paper, we first derive a time-dependent Maxwell’s equation model for simulating wave propagation in anisotropic dispersive media and hyperbolic metamaterials. The modeling equations are obtained by using the Drude–Lorentz model to approximate both the permittivity and permeability. Then we develop a

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1394-1419, June 2024. Abstract. In this paper, we analyze a hybridizable discontinuous Galerkin method for the Helmholtz equation with large wave number, which uses piecewise polynomials of degree of [math] to approximate the potential [math] and its traces and piecewise polynomials of degree of [math] for the flux [math]. It is proved that

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1372-1393, June 2024. Abstract. This paper is concerned with the multi-frequency factorization method for imaging the support of a wave-number-dependent source function. It is supposed that the source function is given by the inverse Fourier transform of some time-dependent source with a priori given radiating period. Using the multi-frequency

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1344-1371, June 2024. Abstract. We adopt the integral definition of the fractional Laplace operator and study an optimal control problem on Lipschitz domains that involves a fractional elliptic PDE as the state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1278-1312, June 2024. Abstract. We consider the problem of numerically approximating the solutions to a partial differential equation (PDE) when there is insufficient information to determine a unique solution. Our main example is the Poisson boundary value problem, when the boundary data is unknown and instead one observes finitely many

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1313-1343, June 2024. Abstract. In this work, we study discrete minimizers of the Ginzburg–Landau energy in finite element spaces. Special focus is given to the influence of the Ginzburg–Landau parameter [math]. This parameter is of physical interest as large values can trigger the appearance of vortex lattices. Since the vortices have to

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1259-1277, June 2024. Abstract. We generalize Bernoulli’s classical method for finding poles of rational functions using the rational orthogonal Malmquist–Takenaka system. We show that our approach overcomes the limitations of previous methods, especially their dependence on the existence of a so-called dominant pole, while significantly

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1226-1258, June 2024. Abstract. We provide a framework to analyze the convergence of discretized kinetic Langevin dynamics for [math]-[math]Lipschitz, [math]-convex potentials. Our approach gives convergence rates of [math], with explicit step size restrictions, which are of the same order as the stability threshold for Gaussian targets and

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1212-1225, June 2024. Abstract. Let [math] be a convex polytope ([math]). The Ritz projection is the best approximation, in the [math]-norm, to a given function in a finite element space. When such finite element spaces are constructed on the basis of quasiuniform triangulations, we show a pointwise estimate on the Ritz projection. Namely

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1191-1211, June 2024. Abstract. We show that generalized multiquadric radial basis functions (RBFs) on [math] have a mean dimension that is [math] as [math] with an explicit bound for the implied constant, under moment conditions on their inputs. Under weaker moment conditions the mean dimension still approaches 1. As a consequence, these

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1171-1190, June 2024. Abstract. We prove a new numerical approximation result for the solutions of semilinear parabolic stochastic partial differential equations, driven by additive space-time white noise in dimension 1. The temporal discretization is performed using an accelerated exponential Euler scheme, and we show that, under appropriate

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1145-1170, June 2024. Abstract.We consider two approximation schemes of the one-dimensional supercooled Stefan problem and prove their convergence, even in the presence of finite time blow-ups. All proofs are based on a probabilistic reformulation recently considered in the literature. The first scheme is a version of the time-stepping scheme

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1119-1144, June 2024. Abstract. This paper considers numerical discretization of a nonlocal conservation law modeling vehicular traffic flows involving nonlocal intervehicle interactions. The nonlocal model involves an integral over the range measured by a horizon parameter and it recovers the local Lighthill–Richards–Whitham model as the

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1098-1118, June 2024. Abstract. Data sites selected from modeling high-dimensional problems often appear scattered in nonpaternalistic ways. Except for sporadic-clustering at some spots, they become relatively far apart as the dimension of the ambient space grows. These features defy any theoretical treatment that requires local or global

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1067-1097, June 2024. Abstract. A key property of the linear Boltzmann semiconductor model is that as the collision frequency tends to infinity, the phase space density [math] converges to an isotropic function [math], called the drift-diffusion limit, where [math] is a Maxwellian and the physical density [math] satisfies a second-order parabolic

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1039-1066, June 2024. Abstract. In this paper, a novel mixed spectral-Galerkin method is proposed and studied for a Maxwell transmission eigenvalue problem in a spherical domain. The method utilizes vector spherical harmonics to achieve dimension reduction. By introducing an auxiliary vector function, the original problem is rewritten as

SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1021-1038, June 2024. Abstract. Randomized quasi-Monte Carlo, via certain scramblings of digital nets, produces unbiased estimates of [math] with a variance that is [math] for any [math]. It also satisfies some nonasymptotic bounds where the variance is no larger than some [math] times the ordinary Monte Carlo variance. For scrambled Sobol’

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 998-1019, April 2024. Abstract. In this paper, we propose a two-level block preconditioned Jacobi–Davidson (BPJD) method for efficiently solving discrete eigenvalue problems resulting from finite element approximations of [math]th ([math]) order symmetric elliptic eigenvalue problems. Our method works effectively to compute the first several

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 974-997, April 2024. Abstract. Accurate evaluation of nearly singular integrals plays an important role in many boundary integral equation based numerical methods. In this paper, we propose a variant of singularity swapping method to accurately evaluate the layer potentials for arbitrarily close targets. Our method is based on the global

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 946-973, April 2024. Abstract. We introduce a predictor-corrector discretization scheme for the numerical integration of a class of stochastic differential equations and prove that it converges with weak order 1.0. The key feature of the new scheme is that it builds up sequentially (and recursively) in the dimension of the state space of

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 919-945, April 2024. Abstract. A general a posteriori error analysis applies to five lowest-order finite element methods for two fourth-order semilinear problems with trilinear nonlinearity and a general source. A quasi-optimal smoother extends the source term to the discrete trial space and, more important, modifies the trilinear term in

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 893-918, April 2024. Abstract. In this note, we design a cut finite element method for a low order divergence-free element applied to a boundary value problem subject to Stokes’ equations. For the imposition of Dirichlet boundary conditions, we consider either Nitsche’s method or a stabilized Lagrange multiplier method. In both cases, the

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 872-892, April 2024. Abstract. Partial differential equations (PDEs) with uncertain or random inputs have been considered in many studies of uncertainty quantification. In forward uncertainty quantification, one is interested in analyzing the stochastic response of the PDE subject to input uncertainty, which usually involves solving high-dimensional

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 842-871, April 2024. Abstract. In this paper, we study the approximability of high-dimensional functions that appear, for example, in the context of many body expansions and high-dimensional model representation. Such functions, though high-dimensional, can be represented as finite sums of lower-dimensional functions. We will derive sampling

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 811-841, April 2024. Abstract. Physics informed neural networks (PINNs) require regularity of solutions of the underlying PDE to guarantee accurate approximation. Consequently, they may fail at approximating discontinuous solutions of PDEs such as nonlinear hyperbolic equations. To ameliorate this, we propose a novel variant of PINNs, termed

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 775-810, April 2024. Abstract. Cell average decomposition (CAD) plays a critical role in constructing bound-preserving (BP) high-order discontinuous Galerkin and finite volume methods for hyperbolic conservation laws. Seeking optimal CAD (OCAD) that attains the mildest BP Courant–Friedrichs–Lewy (CFL) condition is a fundamentally important

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 749-774, April 2024. Abstract. We consider a Beckmann formulation of an unbalanced optimal transport (UOT) problem. The [math]-convergence of this formulation of UOT to the corresponding optimal transport (OT) problem is established as the balancing parameter [math] goes to infinity. The discretization of the problem is further shown to be

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 718-748, April 2024. Abstract. At the fully discrete setting, stability of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions requires local test spaces that ensure the existence of Fortin operators. We construct such operators for [math] and [math] on simplices in any space dimension and arbitrary polynomial degree

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 692-717, April 2024. Abstract. We construct a right inverse of the trace operator [math] on the reference triangle [math] that maps suitable piecewise polynomial data on [math] into polynomials of the same degree and is bounded in all [math] norms with [math] and [math]. The analysis relies on new stability estimates for three classes of

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 667-691, April 2024. Abstract. We study the convergences of three projected Sobolev gradient flows to the ground state of the Gross–Pitaevskii eigenvalue problem. They are constructed as the gradient flows of the Gross–Pitaevskii energy functional with respect to the [math]-metric and two other equivalent metrics on [math], including the