SIAM Journal on Numerical Analysis, Volume 60, Issue 4, Page 2046-2068, January 2022. Many nonlocal models have adopted Euclidean balls as the nonlocal interaction neighborhoods. When solving them numerically, it is sometimes convenient to adopt polygonal approximations of such balls. A crucial question is to what extent such approximations affect the nonlocal operators and the corresponding solutions

SIAM Journal on Numerical Analysis, Volume 60, Issue 4, Page 2014-2045, January 2022. Identifying hidden dynamics from observed data is a significant and challenging task in a wide range of applications. Recently, the combination of linear multistep methods (LMMs) and deep learning has been successfully employed to discover dynamics, whereas a complete convergence analysis of this approach is still

SIAM Journal on Numerical Analysis, Volume 60, Issue 4, Page 1985-2013, August 2022. Numerically solving the 2D Helmholtz equation is widely known to be very difficult largely due to its highly oscillatory solution, which brings about the pollution effect. A very fine mesh size is necessary to deal with a large wavenumber leading to a severely ill-conditioned huge coefficient matrix. To understand

SIAM Journal on Numerical Analysis, Volume 60, Issue 4, Page 1962-1984, August 2022. We establish improved uniform error bounds on a second-order Strang time-splitting method which is equivalent to an exponential wave integrator for the long-time dynamics of the nonlinear Klein--Gordon equation (NKGE) with weak cubic nonlinearity, whose strength is characterized by $\varepsilon^2$ with $0 < \varepsilon

SIAM Journal on Numerical Analysis, Volume 60, Issue 4, Page 1932-1961, August 2022. Several div-conforming and divdiv-conforming finite elements for symmetric tensors on simplexes in arbitrary dimension are constructed in this work. The shape function space is first split as the trace space and the bubble space. The later is further decomposed into the null space of the differential operator and its

SIAM Journal on Numerical Analysis, Volume 60, Issue 4, Page 1905-1931, August 2022. The energy dissipation law and the maximum bound principle (MBP) are two important physical features of the well-known Allen--Cahn equation. While some commonly used first-order time stepping schemes have turned out to preserve unconditionally both the energy dissipation law and the MBP for the equation, restrictions

SIAM Journal on Numerical Analysis, Volume 60, Issue 4, Page 1879-1904, August 2022. Fractional Gaussian noise models the time series with long-range dependence; when the Hurst index $H\in(1/2,1)$, it has positive correlation reflecting a persistent autocorrelation structure. This paper studies the numerical method for solving the stochastic fractional diffusion equation driven by fractional Gaussian

SIAM Journal on Numerical Analysis, Volume 60, Issue 4, Page 1853-1878, August 2022. In the present contribution we develop a sharper error analysis for the Virtual Element Method, applied to a model elliptic problem, that separates the element boundary and element interior contributions to the error. As a consequence we are able to propose a variant of the scheme that allows one to take advantage

SIAM Journal on Numerical Analysis, Volume 60, Issue 4, Page 1824-1852, August 2022. In this paper, we analyze two numerical staggered schemes, one fully implicit and the other semi-implicit, for the compressible Navier--Stokes equations with a singular pressure law: a system which degenerates towards the free-congested Navier--Stokes equations with the density constraint $0 \leq \rho \leq 1$ as the

SIAM Journal on Numerical Analysis, Volume 60, Issue 4, Page 1799-1823, August 2022. In this paper, we propose a new family of fully discrete Sinc-$\theta$ schemes for solving backward stochastic differential equations (BSDEs). More precisely, we consider the $\theta$-schemes for the temporal discretizations and then adopt the Sinc approximations to approximate the associated conditional mathematical

SIAM Journal on Numerical Analysis, Volume 60, Issue 4, Page 1774-1798, August 2022. We consider residual-based a posteriori error estimators for Galerkin discretizations of time-harmonic Maxwell's equations. We focus on configurations where the frequency is high, or close to a resonance frequency, and derive reliability and efficiency estimates. In contrast to previous related works, our estimates

SIAM Journal on Numerical Analysis, Volume 60, Issue 4, Page 1741-1773, August 2022. In this paper we shall establish an a priori L$^2$-norm error estimate of the fourth order Runge--Kutta discontinuous Galerkin method for solving sufficiently smooth solutions of one-dimensional scalar nonlinear conservation laws. The optimal order of accuracy in time is obtained under the standard Courant--Friedrichs--Lewy

SIAM Journal on Numerical Analysis, Volume 60, Issue 4, Page 1701-1740, August 2022. We are interested in the Euler--Maruyama discretization of a stochastic differential equation in dimension $d$ with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get rid of any regularity assumption of the drift in this variable

SIAM Journal on Numerical Analysis, Volume 60, Issue 4, Page 1665-1700, August 2022. In this paper, we investigate the problem of reconstructing the historical distribution for a nonlinear diffusion equation, in which the diffusion is driven by not only a nonlocal operator but also a locally one. The problem naturally arises in many real-world applications including the biological population dynamic

SIAM Journal on Numerical Analysis, Volume 60, Issue 4, Page 1631-1664, August 2022. In this paper, we develop a provably energy stable and conservative discontinuous spectral element method for the shifted wave equation in second order form. The proposed method combines the advantages and central ideas of the following very successful numerical techniques: the summation-by-parts finite difference

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1607-1629, June 2022. In this paper, a well-posed simultaneous space-time first order system least squares formulation is constructed of the instationary incompressible Stokes equations with slip boundary conditions. As a consequence of this well-posedness, the minimization over any conforming triple of finite element spaces for velocities

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1574-1606, June 2022. This is the third part in a series on a mass conserving, high order, mixed finite element method for 2D Stokes flow. In this part, we study a block-diagonal preconditioner for the indefinite Schur complement system arising from the discretization of the Stokes equations using these elements. The underlying finite element

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1548-1573, June 2022. Here, we provide a unified framework for numerical analysis of the stochastic nonlinear fractional diffusion equation driven by fractional Gaussian noise with Hurst index $H\in(0,1)$. A novel estimate of the second moment of the stochastic integral with respect to fractional Brownian motion is constructed, which greatly

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1516-1547, June 2022. We develop a stable finite difference method for the elastic wave equation in bounded media, where the material properties can be discontinuous at curved interfaces. The governing equation is discretized in second order form by a fourth or sixth order accurate summation-by-parts operator. The mesh size is determined

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1494-1515, June 2022. The ensemble Kalman filter (EnKF) belongs to the class of iterative particle filtering methods and can be used for solving control-to-observable inverse problems. In this context, the EnKF is known as ensemble Kalman inversion (EKI). In recent years several continuous limits in the number of iterations and particles

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1472-1493, June 2022. In this paper we propose two exponential integrators of first and second order applied to a class of quasilinear wave-type equations. The analytical framework is an extension of the classical Kato framework and covers quasilinear Maxwell's equations in full space and on a smooth domain as well as a class of quasilinear

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1450-1471, June 2022. We consider an elliptic linear-quadratic parameter estimation problem with a finite number of parameters. A novel a priori bound for the parameter error is proved and, based on this bound, an adaptive finite element method driven by an a posteriori error estimator is presented. Unlike prior results in the literature

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1385-1449, June 2022. In this paper, we develop and analyze numerical methods for high-dimensional Fokker--Planck equations by leveraging generative models from deep learning. Our starting point is a formulation of the Fokker--Planck equation as a system of ordinary differential equations (ODEs) on finite-dimensional parameter space with

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1363-1384, June 2022. We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Surprisingly, a certain weighted least squares recovery operator which uses random samples from a tailored distribution leads to near-optimal results in several relevant situations. The results are stated in terms

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1331-1362, June 2022. A framework to construct time-stable finite-difference schemes that apply boundary conditions strongly (or exactly) is presented for hyperbolic systems. A strong time-stability definition that applies to problems with homogeneous as well as nonhomogeneous boundary data is introduced. Sufficient conditions for strong

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1307-1330, June 2022. This paper presents the first family of conforming finite element $\ddiv\ddiv$ complexes on tetrahedral grids in three dimensions. In these complexes, finite element spaces of $H(\ddiv\ddiv,\Omega;\Sbb)$ are from a recent article [L. Chen and X. Huang, Math. Comp., 91 (2022), pp. 1107--1142] while finite element spaces

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1281-1306, June 2022. Adaptive finite element methods (AFEMs) are an increasingly common means of automatically controlling error in numerical simulations. Proofs of convergence and rate of convergence exist for AFEMs; however, these proofs typically rely upon a nested structure for the sequence of meshes. A metric adaptive finite element

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1251-1280, June 2022. The maximum norm error estimations for virtual element methods are studied. To establish the error estimations, we prove higher local regularity based on delicate analysis of Green's functions and high-order local error estimations for the partition of the virtual element solutions. The maximum norm of the exact gradient

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1226-1250, June 2022. This paper considers a Monte-Carlo Nystrom method for solving integral equations of the second kind, whereby the values $(z(y_i))_{1\leq i\leq N}$ of the solution $z$ at a set of $N$ random and independent points $(y_i)_{1\leq i\leq N}$ are approximated by the solution $(z_{N,i})_{1\leq i\leq N}$ of a discrete $N$-dimensional

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1193-1225, June 2022. We investigate a mortar technique for mixed finite element approximations of a class of domain decomposition saddle point problems on nonmatching grids in which the variable associated with the essential boundary condition, referred to as flux, is chosen as the coupling variable. It plays the role of a Lagrange multiplier

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1168-1192, June 2022. We consider the computation of quadrature rules that are exact for a Chebyshev set of linearly independent functions on an interval $[a,b]$. A general theory of Chebyshev sets guarantees the existence of rules with a Gaussian property, in the sense that $2l$ basis functions can be integrated exactly with just $l$ points

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1136-1167, June 2022. In this paper, we study the propagation speeds of reaction-diffusion-advection fronts in time-periodic cellular and chaotic flows with Kolmogorov--Petrovsky--Piskunov (KPP) nonlinearity. We first apply the variational principle to reduce the computation of KPP front speeds to a principal eigenvalue problem of a linear

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1111-1135, June 2022. Minimum residual methods such as the least-squares finite element method (FEM) or the discontinuous Petrov--Galerkin (DPG) method with optimal test functions usually exclude singular data, e.g., non-square-integrable loads. We consider a DPG method and a least-squares FEM for the Poisson problem. For both methods we

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1083-1110, June 2022. This paper proposes and examines a finite element method (FEM) for a Maxwell obstacle problem in electric shielding. The model is given by a coupled system comprising the Faraday equation and an evolutionary variational inequality (VI) of Ampère--Maxwell-type. Based on the leapfrog (Yee) time-stepping and the Nédélec

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1055-1082, June 2022. For first-order discretizations of the integral fractional Laplacian, we provide sharp local error estimates on proper subdomains in both the local $H^1$-norm and the localized energy norm. Our estimates have the form of a local best approximation error plus a global error measured in a weaker norm.

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1026-1054, June 2022. We construct and analyze first- and second-order implicit-explicit schemes based on the scalar auxiliary variable approach for the magneto-hydrodynamic equations. These schemes are linear, only require solving a sequence of linear differential equations with constant coefficients at each time step, and are unconditionally

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 999-1025, June 2022. In this paper, we propose an embedded low-regularity integrator (ELRI) under a new framework for solving the modified Korteweg-de Vries (mKdV) equation under rough data. Different from the previous work [Wu and Zhao, BIT, Number. Math., (2021)], the present ELRI scheme is constructed based on an approximation of a scaled

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 970-998, June 2022. In the second part of this series, we use the Lagrange multiplier approach proposed in the first part [Comput. Methods Appl. Mech. Engr., 391 (2022), 114585] to construct efficient and accurate bound and/or mass preserving schemes for a class of semilinear and quasi-linear parabolic equations. We establish stability results

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 940-969, June 2022. We propose and analyze an optimal mass transport method for a random genetic drift problem driven by a Moran process under weak selection. The continuum limit, formulated as a reaction-advection-diffusion equation known as the Kimura equation, inherits degenerate diffusion from the discrete stochastic process that conveys

SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 913-939, June 2022. We propose and analyze a new parallel paradigm that uses both the time and the space directions. The original approach couples the Parareal algorithm with incomplete optimized Schwarz waveform relaxation (OSWR) iterations. The analysis of this coupled method is presented for a one-dimensional advection-reaction-diffusion

SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 888-912, April 2022. We propose a novel class of uniformly accurate integrators for the Klein--Gordon equation which capture classical $c=1$ as well as highly oscillatory nonrelativistic regimes $c\gg1$ and, at the same time, allow for low regularity approximations. In particular, the schemes converge with order $\tau$ and $\tau^2$, respectively

SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 856-887, April 2022. The Ericksen model for nematic liquid crystals couples a director field with a scalar degree of orientation variable and allows the formation of various defects with finite energy. We propose a simple but novel finite element approximation of the problem that can be implemented easily within standard finite element packages

SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 824-855, April 2022. In this paper we deal with pointwise approximation of solutions of stochastic differential equations (SDEs) driven by an infinite dimensional Wiener process, with additional jumps generated by a Poisson random measure. Further investigations contain upper error bounds for the proposed truncated dimension randomized Euler

SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 781-803, April 2022. We present a detailed convergence analysis for an operator splitting scheme proposed in [C. Liu, C. Wang, and Y. Wang, J. Comput. Phys., 436 (2021), 110253] for a reaction-diffusion system with detailed balance. The numerical scheme has been constructed based on a recently developed energetic variational formulation,

SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 804-823, April 2022. The main task of this paper is to prove that the perfectly matched layers (PML) method converges exponentially with respect to the PML parameter for scattering problems with periodic surfaces. In [S. N. Chandler-Wilde and P. Monk, Appl. Numer. Math., 59 (2009), pp. 2131--2154], a linear convergence is proved for the PML

SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 715-750, April 2022. We consider a model of energy minimization arising in the study of the mechanical behavior caused by cell contraction within a fibrous biological medium. The macroscopic model is based on the theory of non rank-one convex nonlinear elasticity for phase transitions. We study appropriate numerical approximations based on

SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 688-714, April 2022. A space-time Trefftz discontinuous Galerkin method for the Schrödinger equation with piecewise-constant potential is proposed and analyzed. Following the spirit of Trefftz methods, trial and test spaces are spanned by nonpolynomial complex wave functions that satisfy the Schrödinger equation locally on each element of

SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 659-687, April 2022. Convergence of an adaptive collocation method for the parametric stationary diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic

SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 631-658, April 2022. We present a novel monolithic divergence-conforming HDG scheme for a linear fluid-structure interaction problem with a thick structure. A pressure-robust optimal energy-norm estimate is obtained for the semidiscrete scheme. When combined with a Crank--Nicolson time discretization, our fully discrete scheme is energy stable

SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 606-630, April 2022. We continue the analysis of nonlinear conservation laws on networks initiated in [M. Musch, U. S. Fjordholm, and N. H. Risebro, Netw. Heterog. Media, 17 (2022), pp. 101--128] by extending our analysis to a large class of flux functions which must be neither monotone nor convex/concave. We utilize the framework laid down

SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 585-605, April 2022. We prove convergence for the nonoverlapping Robin--Robin method applied to nonlinear elliptic equations with a $p$-structure, including degenerate diffusion equations governed by the $p$-Laplacian. This nonoverlapping domain decomposition is commonly encountered when discretizing elliptic equations, as it enables the

SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 558-584, April 2022. We conduct a detailed study of the fully discrete finite element approximation of the data completion problem. This is the continuation of [Numer. Math., 139 (2016), pp. 1--25], where the variational problem, resulting from the Kohn--Vogelius duplication framed into the Steklov--Poincaré condensation approach, was semidiscretized

SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 529-557, April 2022. This work is dedicated to a refined error analysis of the high-order transparent boundary conditions for the weighted wave equation on a fractal tree, introduced in the companion work [P. Joly and M. Kachanovska, SIAM J. Sci. Comput., 43 (2021), pp. A3760--A3788]. The construction of such boundary conditions relies on

SIAM Journal on Numerical Analysis, Volume 60, Issue 2, Page 503-528, April 2022. An exponential type of convolution quadrature is proposed as a time-stepping method for the nonlinear subdiffusion equation with bounded measurable initial data. The method combines contour integral representation of the solution, quadrature approximation of contour integrals, multistep exponential integrators for ordinary

SIAM Journal on Numerical Analysis, Volume 60, Issue 1, Page 475-501, February 2022. We derive a residual based a posteriori error estimate for the outer normal flux of approximations to the diffusion problem with variable coefficient. By analyzing the solution of the adjoint problem, we show that error indicators in the bulk may be defined to be of higher order than those close to the boundary, which

SIAM Journal on Numerical Analysis, Volume 60, Issue 1, Page 450-474, February 2022. We study Dirichlet boundary control of Stokes flows in 2D polygonal domains. We consider cost functionals with two different boundary control regularization terms: the ${$L$}^2(\Gamma)$-norm and an energy space seminorm. We prove well-posedness, provide first order optimality conditions, derive regularity results,

SIAM Journal on Numerical Analysis, Volume 60, Issue 1, Page 423-449, February 2022. In this work we present a class of high order unconditionally strong stability preserving (SSP) implicit two-derivative Runge--Kutta schemes and SSP implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes where the time-step restriction is independent of the stiff term. The unconditional SSP property for a method

SIAM Journal on Numerical Analysis, Volume 60, Issue 1, Page 396-422, February 2022. We develop an algorithm that computes strongly continuous semigroups on infinite-dimensional Hilbert spaces with explicit error control. Given a generator $A$, a time $t>0$, an arbitrary initial vector $u_0$, and an error tolerance $\epsilon>0$, the algorithm computes $\exp(tA)u_0$ with error bounded by $\epsilon$

SIAM Journal on Numerical Analysis, Volume 60, Issue 1, Page 364-395, February 2022. Cross-diffusion systems are systems of nonlinear parabolic partial differential equations that are used to describe dynamical processes in several application, including chemical concentrations and cell biology. We present a space-time approach to the proof of existence of bounded weak solutions of cross-diffusion