SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2059-2078, January 2020. Probabilistic numerics aims to study numerical algorithms from a stochastic perspective. This field has recently evolved into a surging interdisciplinary research area (between numerical approximation and probability theory) attracting much attention from the data science community at large. Motivated by this development

SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2019-2058, January 2020. In this paper, we propose a new stabilized projection-based proper orthogonal decomposition reduced order model (POD-ROM) for the numerical simulation of incompressible flows. The new method draws inspiration from successful numerical stabilization techniques used in the context of finite element (FE) methods, such

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 1367-1391, January 2020. We propose a framework that allows us to analyze different variants of hybridizable discontinuous Galerkin (HDG) methods for the static Maxwell equations using one simple analysis. It reduces all the work to the construction of projections that best fit the structures of the approximation spaces. As applications,

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 1339-1366, January 2020. In this paper, we consider the Galerkin finite element methods for the Maxwell--Klein--Gordon system in the Coulomb gauge. We propose a semidiscrete finite element method for the system with the mixed finite element approximation of the vector potential. Energy conservation and error estimates are established for

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 1319-1338, January 2020. A time-stepping $L1$ scheme for subdiffusion equation with a Riemann--Liouville time fractional derivative is developed and analyzed. This is the first paper to show that the $L1$ scheme for the model problem under consideration is second-order accurate (sharp error estimate) over nonuniform time steps. The established

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 1295-1318, January 2020. The singular values and singular vectors of a compact operator $T$ can be estimated by discretizing $T$ (in a variety of ways) and then computing the singular value decomposition of a suitably scaled Galerkin matrix. In general, the singular values and singular vectors converge at the same rate, which is governed

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 1263-1294, January 2020. Ensemble Kalman inversion is a parallelizable methodology for solving inverse or parameter estimation problems. Although it is based on ideas from Kalman filtering, it may be viewed as a derivative-free optimization method. In its most basic form it regularizes ill-posed inverse problems through the subspace property:

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1993-2018, January 2020. We introduce and analyze a sparse spectral method for the solution of Volterra integral equations using bivariate orthogonal polynomials on a triangle domain. The sparsity of the Volterra operator on a weighted Jacobi basis is used to achieve high efficiency and exponential convergence. The discussion is followed

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1965-1992, January 2020. The gradient discretization method (GDM) is a generic framework for designing and analyzing numerical schemes for diffusion models. In this paper, we study the GDM for the porous medium equation, including fast diffusion and slow diffusion models, and a concentration-dependent diffusion tensor. Using discrete functional

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1941-1964, January 2020. This paper is concerned with the analysis of an adaptive edge element method for solving elliptic $curl$-$curl$ variational inequalities of second kind. We derive a posteriori error estimators based on a special combination of the Moreau--Yosida regularization and Nédélec's edge elements of first family. With the

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1918-1940, January 2020. In this paper, a perfectly matched layer (PML) method is proposed to solve the time-domain electromagnetic scattering problems in three dimensions effectively. The PML problem is defined in a spherical layer and derived by using the Laplace transform and real coordinate stretching in the frequency domain. The well-posedness

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1893-1917, January 2020. The stability and error analysis of a second-order fast approximation are considered for the one-dimensional local and nonlocal diffusion equations in the unbounded spatial domain. We first use the conventional central difference scheme to discretize the local second-order spatial derivative operator and use an asymptotically

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1867-1892, January 2020. We solve the following problem: Given piecewise polynomial boundary data $f$ and $g$ on a triangle $K$, satisfying the appropriate compatibility conditions, find an extension $U$ on the triangle that is a polynomial, whose function value and normal derivative on the boundary agree with $f$ and $g$, and such that the

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1845-1866, January 2020. We present a Fourier analysis of wave propagation problems subject to a class of continuous and discontinuous discretizations using high-degree Lagrange polynomials. This allows us to obtain explicit analytical formulas for the dispersion relation and group velocity and, for the first time to our knowledge, characterize

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1822-1844, January 2020. We consider the numerical approximation of single-phase flow in porous media by a mixed finite element method with mass lumping. Our work extends previous results of Wheeler and Yotov, who showed that inexact numerical integration together with an appropriate choice of basis leads to some sort of mass lumping which

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1801-1821, January 2020. A widely used approximation of the Gaussian curvature on a triangulated surface is the angle defect, which measures the deviation between $2\pi$ and the sum of the angles between neighboring edges emanating from a common vertex. We show that the linearization of the angle defect about an arbitrary piecewise constant

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1773-1800, January 2020. We investigate the convergence theory of several known as well as new heuristic parameter choice rules for convex Tikhonov regularization. The success of such methods is dependent on whether certain restrictions on the noise are satisfied. In the linear theory, such conditions are well understood and hold for typically

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1744-1772, January 2020. We propose a new normalized Sobolev gradient flow for the Gross--Pitaevskii eigenvalue problem based on an energy inner product that depends on time through the density of the flow itself. The gradient flow is well-defined and converges to an eigenfunction. For ground states we can quantify the convergence speed as

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1719-1743, January 2020. We propose an iterative algorithm for the numerical computation of sums of squares of polynomials approximating given data at prescribed interpolation points. The method is based on the definition of a convex functional $G$ arising from the dualization of a quadratic regression over the Cholesky factors of the sum

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1696-1718, January 2020. Recently, the $H(div)$-conforming finite element families for second-order elliptic problems have come more into focus since, due to hybridization and subsequent advances in computational efficiency, their use is no longer mainly theoretical. Their property of yielding exactly divergence-free solutions for mixed problems

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1674-1695, January 2020. We propose numerical schemes for a class of Keller--Segel equations. The discretization is based on the gradient flow structure. The resulting first-order scheme is mass conservative, bound preserving, uniquely solvable, and energy dissipative, and the second-order scheme satisfies the first three properties. For

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1654-1673, January 2020. We present a dual finite element method for a singularly perturbed reaction-diffusion problem. It can be considered a reduced version of the mixed finite element method for approximate solutions. The new method only approximates the dual variables without approximating the primary variable. An approximation for the

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1613-1653, January 2020. In two recent publications [M. Kovács, S. Larsson, and A. Mesforush, SIAM J. Numer. Anal., 49 (2011), pp. 2407--2429] and [D. Furihata, et al., SIAM J. Numer. Anal., 56 (2018), pp. 708--731], strong convergence of the semidiscrete and fully discrete finite element methods are, respectively, proved for the Cahn--Hilliard--Cook

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1592-1612, January 2020. We study the Lax consistency of a staggered finite volume scheme for Lagrangian hydrodynamics adapting the methodology introduced by Després in [Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 2669--2679]. The scheme is conservative in mass, total energy, and momentum, and the positivity of the density and internal

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1556-1591, January 2020. We deal with the finite element tearing and interconnecting dual primal preconditioner for elliptic problems discretized by the virtual element method. We extend the result of [S. Bertoluzza, M. Pennacchio, and D. Prada, Calcolo, 54 (2017), pp. 1565--1593] to the three dimensional case. We prove polylogarithmic condition

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1531-1555, January 2020. The main goal of the paper is to show new stability and localization results for the finite element solution of the Stokes system in $W^{1,\infty}$ and $L^\infty$ norms under standard assumptions on the finite element spaces on quasi-uniform meshes in two and three dimensions. Although interior error estimates are

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1495-1530, January 2020. This paper gives a unified convergence analysis of additive Schwarz methods for general convex optimization problems. Resembling the fact that additive Schwarz methods for linear problems are preconditioned Richardson methods, we prove that additive Schwarz methods for general convex optimization are in fact gradient

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1469-1494, January 2020. We consider parametrized problems driven by spatially nonlocal integral operators with parameter-dependent kernels. In particular, kernels with varying nonlocal interaction radius $\delta > 0$ and fractional Laplace kernels, parametrized by the fractional power $s\in(0,1)$, are studied. In order to provide an efficient

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1440-1468, January 2020. In this paper, we first derive the multipole expansion (ME) and local expansion (LE) for far fields from wave sources in two-dimensional (2-D) layered media as well as the multipole-to-local translation (M2L) operator, by using the generating function of Bessel functions and Sommerfeld integral representations of

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1422-1439, January 2020. In this paper, we develop a uniform robust weak Galerkin finite element scheme for Brinkman equations. The major idea for achieving uniform energy-error estimate is to use a divergence preserving velocity reconstruction operator in the discretization. The optimal convergence results for velocity and pressure have

SIAM Journal on Numerical Analysis, Volume 58, Issue 3, Page 1393-1421, January 2020. In this paper, we will present a strong (or pathwise) approximation of standard Brownian motion by a class of orthogonal polynomials. The coefficients that are obtained from the expansion of Brownian motion in this polynomial basis are independent Gaussian random variables. Therefore, it is practical (i.e., requires

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 1239-1262, January 2020. An operator analogue of the FEAST matrix eigensolver is developed to compute the discrete part of the spectrum of a differential operator in a region of interest in the complex plane. Unbounded search regions are handled with a novel rational filter for the right half-plane. If the differential operator is normal

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 1217-1238, January 2020. An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behavior at an initial time. For this problem, we give a simple and general numerical-stability analysis using barrier functions, which yields sharp pointwise-in-time

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 1195-1216, January 2020. We consider the mean dimension of some ridge functions of spherical Gaussian random vectors of dimension $d$. If the ridge function is Lipschitz continuous, then the mean dimension remains bounded as $d\to\infty$. If, instead, the ridge function is discontinuous, then the mean dimension depends on a measure of the

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 1164-1194, January 2020. In [Commun. Pure Appl. Math., 2 (1949), pp. 331--407], Grad proposed a Hermite series expansion for approximating solutions to kinetic equations that have an unbounded velocity space. However, for initial boundary value problems, poorly imposed boundary conditions lead to instabilities in Grad's Hermite expansion

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 1138-1163, January 2020. In this paper, we present an efficient and robust approach to compute a normalized B-spline-like basis for spline spaces with pieces drawn from extended Tchebycheff spaces. The extended Tchebycheff spaces and their dimensions are allowed to change from interval to interval. The approach works by constructing a matrix

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 1117-1137, January 2020. For the numerical solution of parabolic problems with a linear diffusion term, linearly implicit time integrators are considered. To reduce the cost on the linear algebra level, an alternating direction implicit approach is applied (so-called AMF-W-methods). The present work proves optimal bounds of the global error

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 1092-1116, January 2020. For a four-stream approximation of the kinetic model of radiative transfer with isotropic scattering, a numerical scheme endowed with both truly 2D well-balanced and diffusive asymptotic-preserving properties is derived, in the same spirit as what was done in [L. Gosse and G. Toscani, C. R. Math. Acad. Sci. Paris

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 1068-1091, January 2020. Many studies in uncertainty quantification have been carried out under the assumption of an input random field in which a countable number of independent random variables are each uniformly distributed on an interval, with these random variables entering linearly in the input random field (the so-called affine model)

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 1029-1067, January 2020. This work proposes a novel multiscale finite element method for acoustic wave propagation in highly heterogeneous media which is accurate on coarse meshes. It originates from the primal hybridization of the Helmholtz equation at the continuous level, which relaxes the continuity of the unknown on the skeleton of a

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 1008-1028, January 2020. We propose a new fictitious domain finite element method, well suited for elliptic problems posed in a domain given by a level-set function without requiring a mesh fitting the boundary. To impose the Dirichlet boundary conditions, we search the approximation to the solution as a product of a finite element function

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 988-1007, January 2020. We proposed ways to implement meshless collocation methods extrinsically for solving elliptic PDEs on smooth, closed, connected, and complete Riemannian manifolds with arbitrary codimensions. Our methods are based on strong-form collocations with oversampling and least-squares minimizations, which can be implemented

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 962-987, January 2020. We consider a scalar conservation law with hysteresis in which the flux function depends on the history of the evolution. The work is motivated by the need to account for hysteresis in important applications to transport with adsorption. To model hysteresis we use an auxiliary system of ODEs with constraints known as

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 929-961, January 2020. Parabolic partial differential equations (PDEs) and backward stochastic differential equations have a wide range of applications. In particular, high-dimensional PDEs with gradient-dependent nonlinearities appear often in the state-of-the-art pricing and hedging of financial derivatives. In this article we prove that

SIAM Journal on Numerical Analysis, Volume 58, Issue 2, Page 907-928, January 2020. Diagonal norm finite difference based time integration methods in summation-by-parts form are investigated. The second, fourth, and sixth order accurate discretizations are proven to have eigenvalues with strictly positive real parts. This leads to provably invertible fully discrete approximations of initial boundary

SIAM Journal on Numerical Analysis, Volume 58, Issue 1, Page 884-906, January 2020. This paper is devoted to the construction and analysis of the finite element approximations for the $H(D)$ convection-diffusion problems, where $D$ can be chosen as ${grad}$, ${curl}$, or ${div}$ in the three dimension (3D) case. An essential feature of these constructions is to properly average the PDE coefficients

SIAM Journal on Numerical Analysis, Volume 58, Issue 1, Page 858-883, January 2020. In this paper, we focus on the constrained sparse regression problem, where the loss function is convex but nonsmooth and the penalty term is defined by the cardinality function. First, we give an exact continuous relaxation problem in the sense that both problems have the same optimal solution set. Moreover, we show

SIAM Journal on Numerical Analysis, Volume 58, Issue 1, Page 834-857, January 2020. We consider first-kind weakly singular and hypersingular boundary integral operators for the Laplacian on screens in $\mathbb{R}^{3}$ and their Galerkin discretization by means of low-order piecewise polynomial boundary elements. For the resulting linear systems of equations we propose novel Calderón-type preconditioners

SIAM Journal on Numerical Analysis, Volume 58, Issue 1, Page 811-833, January 2020. The article addresses the convergence of implicit and semi-implicit, fully discrete approximations of a class of nonlinear parabolic evolution problems. Such schemes are popular in the numerical solution of evolutions defined with the $p$-Laplace operator since the latter lead to linear systems. The semi-implicit treatment

SIAM Journal on Numerical Analysis, Volume 58, Issue 1, Page 788-810, January 2020. This paper provides theoretical justification that Anderson acceleration (AA) improves the convergence rate of contractive fixed-point iterations in the vicinity of a fixed-point. AA has been used for decades to speed up nonlinear solvers in many applications. However, a rigorous mathematical justification of the improved

SIAM Journal on Numerical Analysis, Volume 58, Issue 1, Page 757-787, January 2020. In the classical theory of fluid mechanics a linear relationship between the shear stress and the symmetric velocity gradient tensor is often assumed. Even when a nonlinear relationship is assumed, it is typically formulated in terms of an explicit relation. Implicit constitutive models provide a theoretical framework

SIAM Journal on Numerical Analysis, Volume 58, Issue 1, Page 733-756, January 2020. An optimal mesh size of the sampling region can help to reduce computational burden in practical applications. In this work, we investigate optimal choices of mesh sizes for the identifications of medium obstacles from either the far-field or near-field data in two and three dimensions. The results would have applications

SIAM Journal on Numerical Analysis, Volume 58, Issue 1, Page 706-732, January 2020. We introduce a new discretization of a mixed formulation of the incompressible Stokes equations that includes symmetric viscous stresses. The method is built upon a mass conserving mixed formulation that we recently studied. The improvement in this work is a new method that directly approximates the viscous fluid stress

SIAM Journal on Numerical Analysis, Volume 58, Issue 1, Page 684-705, January 2020. Multilevel quadrature methods for parametric operator equations such as the multilevel (quasi--) Monte Carlo method resemble a sparse tensor product approximation between the spatial variable and the parameter. We employ this fact to reverse the multilevel quadrature method by applying differences of quadrature rules

SIAM Journal on Numerical Analysis, Volume 58, Issue 1, Page 657-683, January 2020. In this paper, we introduce artificial boundary conditions for the linearized Green--Naghdi system of equations. The derivation of such continuous (respectively, discrete) boundary conditions includes the inversion of Laplace transform (respectively, $\mathcal{Z}$-transform) and these boundary conditions are in turn

SIAM Journal on Numerical Analysis, Volume 58, Issue 1, Page 630-656, January 2020. We develop a new multipoint stress mixed finite element method for linear elasticity with weakly enforced stress symmetry on simplicial grids. Motivated by the multipoint flux mixed finite element method for Darcy flow, the method utilizes the lowest order Brezzi--Douglas--Marini finite element spaces for the stress

SIAM Journal on Numerical Analysis, Volume 58, Issue 1, Page 607-629, January 2020. We prove that a class of monotone finite volume schemes for scalar conservation laws with discontinuous flux converge at a rate of sqrtDelta x in L^1, whenever the flux is strictly monotone in u and the spatial dependency of the flux is piecewise constant with finitely many discontinuities. We also present numerical

SIAM Journal on Numerical Analysis, Volume 58, Issue 1, Page 590-606, January 2020. We consider the approximation of Poisson type problems where the source is given by a singular measure and the domain is a convex polygonal or polyhedral domain. First, we prove the well-posedness of the Poisson problem when the source belongs to the dual of a weighted Sobolev space where the weight belongs to the Muckenhoupt

SIAM Journal on Numerical Analysis, Volume 58, Issue 1, Page 565-589, January 2020. We propose a novel artificial compression, reduced order model (AC-ROM) for the numerical simulation of viscous incompressible fluid flows. The new AC-ROM provides approximations not only for velocity but also for pressure, which is needed to calculate forces on bodies in the flow and to connect the simulation parameters

SIAM Journal on Numerical Analysis, Volume 58, Issue 1, Page 541-564, January 2020. Those studying complex domains containing superposed fluid and porous layers will frequently encounter the phenomenon of natural convection. The mathematical model of this problem can be described using a Navier--Stokes/Darcy system coupled with heat equations. The interface conditions for Navier--Stokes/Darcy systems