SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2133-2156, October 2023. Abstract. We develop a unifying framework for interpolatory [math]-optimal reduced-order modeling for a wide class of problems ranging from stationary models to parametric dynamical systems. We first show that the framework naturally covers the well-known interpolatory necessary conditions for [math]-optimal model

SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2106-2132, October 2023. Abstract. This work establishes [math]-norm stability and convergence for an L2 method on general nonuniform meshes when applied to the subdiffusion equation. Under mild constraints on the time step ratio [math], such as [math] for [math], the positive semidefiniteness of a crucial bilinear form associated with the

SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2084-2105, October 2023. Abstract. We introduce an adaptive superconvergent finite element method for a class of mixed formulations to solve partial differential equations involving a diffusion term. It combines a superconvergent postprocessing technique for the primal variable with an adaptive finite element method via residual minimization

SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2062-2083, October 2023. Abstract. Solution methods for the nonlinear PDE of the Rudin–Osher–Fatemi (ROF) and minimum-surface models are fundamental for many modern applications. Many efficient algorithms have been proposed. First-order methods are common. They are popular due to their simplicity and easy implementation. Some second-order

SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2035-2061, October 2023. Abstract. In this paper, we develop a conditional Monte Carlo method for solving PDEs involving an integral fractional Laplacian on any bounded domain in arbitrary dimensions. We first construct the Feynman–Kac representation based on the Green function and Poisson kernel for the fractional Laplacian operator on the

SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2011-2034, October 2023. Abstract. The solution of the nonlinear initial-value problem [math] for [math] with [math], where [math] is the Caputo derivative of order [math] and [math] are positive parameters, is known to exhibit [math] decay as [math]. No corresponding result for any discretization of this problem has previously been proved

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1989-2010, August 2023. Abstract. We propose and analyze a semidiscrete parametric finite element scheme for solving the area-preserving curve shortening flow. The scheme is based on Dziuk’s approach [SIAM J. Numer. Anal., 36 (1999), pp. 1808–1830] for the anisotropic curve shortening flow. We prove that the scheme preserves two fundamental

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1962-1988, August 2023. Abstract. We develop a hybrid spatial discretization for the wave equation in second order form, based on high-order accurate finite difference methods and discontinuous Galerkin methods. The hybridization combines computational efficiency of finite difference methods on Cartesian grids and geometrical flexibility

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1938-1961, August 2023. Abstract. The symmetric interior penalty discontinuous Galerkin method and its version with weighted averages are considered on shape-regular nonconforming meshes with an arbitrarily large number of mesh faces contained in any element face. For this method, residual-type a posteriori error estimates in the maximum

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1918-1937, August 2023. Abstract. In this work, we propose a high-order multiscale method for an elliptic model problem with rough and possibly highly oscillatory coefficients. Convergence rates of higher order are obtained using the regularity of the right-hand side only. Hence, no restrictive assumptions on the coefficient, the domain,

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1885-1917, August 2023. Abstract. The purpose of this paper is to analyze a mixed method for the linear elasticity eigenvalue problem, which approximates numerically the stress, displacement, and rotation, by piecewise [math], [math], and [math]th degree polynomial functions ([math]), respectively. The numerical eigenfunction of stress is

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1858-1884, August 2023. Abstract. In this paper, we consider the time integration of parabolic equations with block implicit methods (BIM). Depending on the size of the block, high-order BIM with [math]-stability are designed without the need of multiple initial guesses. Similar to Runge–Kutta methods, a BIM can be defined by a tableau including

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1835-1857, August 2023. Abstract. Hyperbolic curvature flow is a geometric evolution equation that in the plane can be viewed as the natural hyperbolic analogue of curve shortening flow. It was proposed by Gurtin and Podio-Guidugli [SIAM J. Math. Anal., 22 (1991), pp. 575–586] to model certain wave phenomena in solid-liquid interfaces. We

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1819-1834, August 2023. Abstract. Based on a quantitative version of the inverse function theorem and an appropriate saddle-point formulation, we derive a quasi-optimal error estimate for the finite element approximation of harmonic maps into spheres with a nodal discretization of the unit-length constraint. The error analysis is based on

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1783-1818, August 2023. Abstract. We present a local construction of [math]-conforming piecewise polynomials satisfying a prescribed curl constraint. We start from a piecewise polynomial not contained in the [math] space but satisfying a suitable orthogonality property. The procedure employs minimizations in vertex patches, and the outcome

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1763-1782, August 2023. Abstract. In the present paper, we prove convergence rates for the pressure of the local discontinuous Galerkin approximation, proposed in Part I of the paper [A. Kaltenbach and M. Růžička, SIAM J. Numer. Anal., 61 (2023), pp. 1613–1640], of systems of [math]-Navier-Stokes type and [math]-Stokes type with [math]. The

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1737-1762, August 2023. Abstract. A new approach to solving eigenvalue optimization problems for large structured matrices is proposed and studied. The class of optimization problems considered is related to computing structured pseudospectra and their extremal points, and to structured matrix nearness problems such as computing the distance

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1716-1736, August 2023. Abstract. This paper deals with the expectation of monomials with respect to the stochastic area integral [math] and the increments of two Wiener processes, [math]. In a monomial, if the exponent of one of the Wiener increments or the stochastic area integral is an odd number, then the expectation of the monomial is

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1689-1715, August 2023. Abstract. In this paper, we propose a new exponential-type integrator for solving the gKdV equation under rough data. By introducing new frequency approximation techniques and a key gauge transform, the proposed scheme is explicit, stable, and efficient in practice. The optimal convergence result of the scheme is established

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1641-1663, August 2023. Abstract. In the present paper, we prove convergence rates for the velocity of the local discontinuous Galerkin approximation, proposed in Part I of the paper [A. Kaltenbach and M. Růžička, SIAM J. Numer. Anal., to appear], of systems of [math]-Navier–Stokes type and [math]-Stokes type with [math]. The convergence

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1613-1640, August 2023. Abstract. In the present paper, we propose a local discontinuous Galerkin approximation for fully nonhomogeneous systems of [math]-Navier–Stokes type. On the basis of the primal formulation, we prove well-posedness, stability (a priori estimates), and weak convergence of the method. To this end, we propose a new discontinuous

SIAM Journal on Numerical Analysis, Volume 61, Issue 4, Page 1613-1637, August 2023. Abstract. We prove the convergence of discontinuous Galerkin approximations for the Vlasov–Poisson system written as a hyperbolic system using Hermite polynomials in velocity. To obtain stability properties, we introduce a suitable weighted [math] space, with a time-dependent weight, and first prove global stability

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1609-1612, June 2023. Abstract. The proof of the main theorem in [B. Li, SIAM J. Numer. Anal., 59 (2021), pp. 1592–1617] is corrected. With the corrected proof, the main theorem in this published paper is still valid.

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1585-1608, June 2023. Abstract. Koopman operators globally linearize nonlinear dynamical systems and their spectral information is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. However, Koopman operators are infinite dimensional, and computing their spectral information is a considerable challenge. We

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1546-1584, June 2023. Abstract. In this paper, a generalized finite element (FE) method with optimal local approximation spaces for solving high-frequency heterogeneous Helmholtz problems is systematically studied. The local spaces are built from selected eigenvectors of carefully designed local eigenvalue problems defined on generalized

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1513-1545, June 2023. Abstract. We prove exponential expression rates of deep operator networks (deep ONets) between infinite-dimensional spaces that emulate the coefficient-to-solution map of linear, elliptic, second-order, divergence-form partial differential equations (PDEs). In particular, we consider problems set in [math]-dimensional

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1482-1512, June 2023. Abstract. In this paper, an efficient ensemble domain decomposition algorithm is proposed for fast solving the fully mixed random Stokes–Darcy model with the physically realistic Beavers–Joseph interface conditions. We utilize the Monte Carlo method for the coupled model with random inputs to derive some deterministic

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1449-1481, June 2023. Abstract. We present a new stress/total-pressure formulation for poroelasticity that incorporates the coupling with steady nonlinear diffusion modified by stress. This nonlinear problem is written in mixed-primal form, which combines a perturbed twofold saddle-point system with an elliptic problem. We analyze the continuous

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1426-1448, June 2023. Abstract. The suboptimality of Gauss–Hermite quadrature and the optimality of the trapezoidal rule are proved in the weighted Sobolev spaces of square integrable functions of order [math], where the optimality is in the sense of worst-case error. For Gauss–Hermite quadrature, we obtain matching lower and upper bounds

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1405-1425, June 2023. Abstract. The aim of this paper is to construct and analyze explicit exponential Runge–Kutta methods for the temporal discretization of linear and semilinear integro-differential equations. By expanding the errors of the numerical method in terms of the solution, we derive order conditions that form the basis of our

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1386-1404, June 2023. Abstract. There are plenty of applications and analyses for time-independent elliptic partial differential equations in the literature hinting at the benefits of overtesting by using more collocation conditions than the number of basis functions. Overtesting not only reduces the problem size, but is also known to be

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1369-1385, June 2023. Abstract. For systems of the form [math], [math], common in many applications, we analyze splitting integrators based on the (linear/nonlinear) split systems [math], [math] and [math], [math]. We show that the well-known Strang splitting is optimally stable in the sense that, when applied to a relevant model problem

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1340-1368, June 2023. Abstract. In this paper we study the influence of including snapshots that approach the velocity time derivative in the numerical approximation of the incompressible Navier–Stokes equations by means of proper orthogonal decomposition (POD) methods. Our set of snapshots includes the velocity approximation at the initial

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1316-1339, June 2023. Abstract. In this work, we present a new numerical method for solving the scalar transmission problem with sign-changing coefficients. In electromagnetism, such a transmission problem can occur if the domain of interest is made of a classical dielectric material and a metal or a metamaterial, with, for instance, an electric

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1293-1315, June 2023. Abstract. We prove that the recently developed semiexplicit symplectic integrators for nonseparable Hamiltonian systems preserve any linear and quadratic invariants possessed by the Hamiltonian systems. This is in addition to being symmetric and symplectic as shown in our previous work; hence, it shares the crucial

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1278-1292, June 2023. Abstract. When solving the Helmholtz equation numerically, the accuracy of numerical solution deteriorates as the wave number [math] increases, which is known as “pollution effect” and which is directly related to the phase difference between the exact and numerical solutions, caused by the numerical dispersion. In this

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1246-1277, June 2023. Abstract. In this paper, we consider a class of highly oscillatory Hamiltonian systems which involve a scaling parameter [math]. The problem arises from many physical models in some limit parameter regime or from some time-compressed perturbation problems. The solution of the model exhibits rapid temporal oscillations

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1218-1245, June 2023. Abstract. The paper is concerned with the analysis of a popular convex-splitting finite element method (FEM) for the Cahn–Hilliard–Navier–Stokes system, which has been widely used in practice. Since the method is based on a combined approximation to multiple variables involved in the system, the approximation to one

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1195-1217, June 2023. Abstract. In this paper, we propose a new spectral decomposition method to simulate waves propagating in complicated waveguides. For the numerical solutions of waveguide scattering problems, an important task is to approximate the Dirichlet-to-Neumann (DtN) map efficiently. From previous results, the physical solution

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1172-1194, June 2023. Abstract. In optimal control problems defined on stratified domains, the dynamics and the running cost may have discontinuities on a finite union of submanifolds of [math]. In [A. Bressan and Y. Hong, Netw. Heterog. Media, 2 (2007), pp. 313–331], [G. Barles and E. Chasseigne, Netw. Heterog. Media, 10 (2015), pp. 809–836]

SIAM Journal on Numerical Analysis, Volume 61, Issue 3, Page 1139-1171, June 2023. Abstract. We consider the problem of domain approximation in finite element methods for Maxwell equations on general curved domains, i.e., when affine or polynomial meshes fail to exactly cover the domain of interest and an exact parametrization of the surface may not be readily available. In such cases, one is forced

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 1057-1079, April 2023. Abstract. Preconditioning for Krylov methods often relies on operator theory when mesh independent estimates are looked for. The goal of this paper is to contribute to the long development of the analysis of superlinear convergence of Krylov iterations when the preconditioned operator is a compact perturbation of the

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 1018-1056, April 2023. Abstract. For coupling problems such as the Navier–Stokes–Darcy problem in porous and fluid layers, a natural question is that, while the orders of polynomials approximating the variables are different, could the lower order elements pollute the higher ones? In this paper, the numerical pollution of a decoupled algorithm

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 995-1017, April 2023. Abstract. We present a discontinuous Petrov–Galerkin method with optimal test functions for the Reissner–Mindlin plate bending model. Our method is based on a variational formulation that utilizes a Helmholtz decomposition of the shear force. It produces approximations of the primitive variables and the bending moments

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 973-994, April 2023. Abstract. Optimal control problem is typically solved by first finding the value function through the Hamilton–Jacobi equation (HJE) and then taking the minimizer of the Hamiltonian to obtain the control. In this work, instead of focusing on the value function, we propose a new formulation for the gradient of the value

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 952-972, April 2023. Abstract. Chebyshev spectral methods are widely used in numerical computations. When the underlying function has a singularity, it was observed by Trefethen in 2011 that its Chebyshev interpolants exhibit an error localization property, that is, their errors in a neighborhood of the singularity are obviously larger than

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 929-951, April 2023. Abstract. This paper is concerned with new error analysis of a lowest-order backward Euler Galerkin-mixed finite element method for the time-dependent Ginzburg–Landau equations. The method is based on a commonly-used nonuniform approximation, in which a linear Lagrange element, the lowest-order Nédélec edge element, and

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 837-871, April 2023. Abstract. In this paper, we present, analyze, and test a novel low-complexity time-stepping finite element method for natural convection problems utilizing a time filter (TF). First, via a TF to postprocess the solutions of backward Euler (BE) schemes, we make a minimally intrusive modification to the existing codes to

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 905-928, April 2023. Abstract. Existence and connection of numerical attractors for discrete-time [math]-Laplace lattice systems via the implicit Euler scheme are proved. The numerical attractors are shown to have an optimized bound, which leads to the continuous convergence of the numerical attractors when the graph of the nonlinearity closes

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 872-904, April 2023. Abstract. Strong stability (or monotonicity)-preserving time discretization schemes preserve the stability properties of the exact solution and have proved very useful in scientific and engineering computation, especially in solving hyperbolic partial differential equations. The main aim of this work is to further extend

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 812-836, April 2023. Abstract. An extra-stabilized Morley finite element method (FEM) directly computes guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplacian Dirichlet eigenvalues. The smallness assumption [math] [math] in 2D (resp., [math] in 3D) on the maximal mesh-size [math] makes the computed

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 784-811, April 2023. Abstract. In this paper we discretize the incompressible Navier–Stokes equations in the framework of finite element exterior calculus. We make use of the Lamb identity to rewrite the equations into a vorticity-velocity-pressure form which fits into the de Rham complex of minimal regularity. We propose a discretization

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 1080-1102, April 2023. Abstract. A kernel method for estimating a probability density function from an independent and identically distributed sample drawn from such density is presented. Our estimator is a linear combination of kernel functions, the coefficients of which are determined by a linear equation. An error analysis for the mean

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 1103-1137, April 2023. Abstract. A fully discrete finite difference scheme for stochastic reaction-diffusion equations driven by a [math]-dimensional white noise is studied. The optimal strong rate of convergence is proved without posing any regularity assumption on the nonlinear reaction term. The proof relies on stochastic sewing techniques

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 755-783, April 2023. Abstract. We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems, where the nonlinear reaction terms need to be neither monotone nor convex. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy reduction approach. More

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 733-754, April 2023. Abstract. Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 707-732, April 2023. Abstract. We consider a fluid-fluid interaction problem, where two flows (with high Reynolds numbers for one or both of these flows) are coupled through a joint interface. A nonlinear coupling equation, known as the rigid lid condition, creates an extra level of difficulty, typical for atmosphere-ocean problems. We propose

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 675-706, April 2023. Abstract. We develop a space-time mortar mixed finite element method for parabolic problems. The domain is decomposed into a union of subdomains discretized with nonmatching spatial grids and asynchronous time steps. The method is based on a space-time variational formulation that couples mixed finite elements in space

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 642-674, April 2023. Abstract. In this paper, based on a domain decomposition method, we shall propose a two-level preconditioned Helmholtz subspace iterative (PHSI) method for solving algebraic eigenvalue problems resulting from edge element approximation of Maxwell eigenvalue problems. The two-level PHSI method may compute simple eigenpairs

SIAM Journal on Numerical Analysis, Volume 61, Issue 2, Page 617-641, April 2023. Abstract. We deal with a long-standing problem about how to design an energy-stable numerical scheme for solving the motion of a closed curve under anisotropic surface diffusion with a general anisotropic surface energy [math] in two dimensions, where [math] is the outward unit normal vector. By introducing a novel surface