SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3406-3426, January 2020. This work presents a comprehensive discretization theory for abstract linear operator equations in Banach spaces. The fundamental starting point of the theory is the idea of residual minimization in dual norms and its inexact version using discrete dual norms. It is shown that this development, in the case of strictly

SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3382-3405, January 2020. Many important initial value problems have the property that energy is nonincreasing in time. Energy stable methods, also referred to as strongly stable methods, guarantee the same property discretely. We investigate requirements for conditional energy stability of explicit Runge--Kutta methods for nonlinear or nonautonomous

SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3355-3381, January 2020. Reduced model spaces, such as reduced bases and polynomial chaos, are linear spaces $V_n$ of finite dimension $n$ which are designed for the efficient approximation of certain families of parametrized PDEs in a Hilbert space $V$. The manifold $\cal M$ that gathers the solutions of the PDE for all admissible parameter

SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3332-3354, January 2020. Using dimensionally reduced models for the numerical simulation of thin objects is highly attractive as this reduces the computational work substantially. The case of narrow thin elastic bands is considered, and a convergent finite element discretization for the one-dimensional energy functional and a fully practical

SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3309-3331, January 2020. In this paper we analyze the finite element approximation of the Stokes equations with nonsmooth Dirichlet boundary data. To define the discrete solution, we first approximate the boundary datum by a smooth one and then apply a standard finite element method to the regularized problem. We prove almost optimal order

SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3286-3308, January 2020. This paper studies adaptive first-order system least-squares finite element methods (LSFEMs) for second-order elliptic partial differential equations in nondivergence form. Unlike the classical finite element methods which use weak formulations of PDEs that are not applicable for the nondivergence equation, the first-order

SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3251-3285, January 2020. In our previous work [J. R. Singler, SIAM J. Numer. Anal., 52 (2014), pp. 852--876], we considered the proper orthogonal decomposition (POD) of time varying PDE solution data taking values in two different Hilbert spaces. We considered various POD projections of the data and obtained new results concerning POD projection

SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3226-3250, January 2020. The aim of this paper is to develop and analyze high-order time stepping schemes for approximately solving semilinear subdiffusion equations. We apply the convolution quadrature generated by $k$-step backward differentiation formula (BDF$k$) to discretize the time-fractional derivative with order $\alpha\in (0,1)$

SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3197-3225, January 2020. High order methods are often desired for the evolution of ordinary differential equations, in particular those arising from the semidiscretization of partial differential equations. In prior work we investigated the interplay between the local truncation error and the global error to construct error inhibiting general

SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3165-3196, January 2020. We present a semi-Lagrangian scheme for the approximation of a class of Hamilton--Jacobi--Bellman (HJB) equations on networks. The scheme is explicit, consistent, and stable for large time steps. We prove a convergence result and two error estimates. For an HJB equation with space-independent Hamiltonian, we obtain

SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3144-3164, January 2020. We show that the expected solution operator of prototypical linear elliptic PDEs with random coefficients is well approximated by a computable sparse matrix. This result is based on a random localized orthogonal multiresolution decomposition of the solution space that allows both the sparse approximate inversion of

SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3125-3143, January 2020. Parametric sensitivity analysis is a critical component in the study of mathematical models of physical systems. Due to its simplicity, finite difference methods are used extensively for this analysis in the study of stochastically modeled reaction networks. Different coupling methods have been proposed to build finite

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 3091-3123, January 2020. In this paper we are concerned with numerical methods for Helmholtz equations with large wave numbers. We design a least squares method for discretization of the considered Helmholtz equations. In this method, an auxiliary unknown is introduced on the common interface of any two neighboring elements and a quadratic

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 3068-3090, January 2020. We propose a new family of models for fluid flows at high Reynolds numbers, large eddy simulation with correction (LES-C), that combines a LES approach to turbulence modeling with a defect correction methodology. We investigate, both numerically and theoretically, one of these models, based on the approximate deconvolution

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 3040-3067, January 2020. In this paper, we develop efficient stochastic structure-preserving schemes to compute the effective diffusivity for particles moving in random flows. We first introduce the motion of a passive tracer particle in random flows using the Lagrangian formulation, which is modeled by stochastic differential equations (SDEs)

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 3010-3039, January 2020. We analyze the convergence of piecewise collocation methods for computing periodic solutions of general retarded functional differential equations under the abstract framework recently developed in [S. Maset, Numer. Math., 133 (2016), pp. 525--555], [S. Maset, SIAM J. Numer. Anal., 53 (2015), pp. 2771--2793], and

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2981-3009, January 2020. We present a new parareal algorithm based on a diagonalization technique proposed recently. The algorithm uses a single implicit Runge--Kutta method with the same small step-size for both the $\mathcal{F}$ and $\mathcal{G}$ propagators in parareal and would thus converge in one iteration when used directly like this

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2953-2980, January 2020. The periodization of a stationary Gaussian random field on a sufficiently large torus comprising the spatial domain of interest is the basis of various efficient computational methods, such as the classical circulant embedding technique using the fast Fourier transform for generating samples on uniform grids. For

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2934-2952, January 2020. This paper develops an efficient numerical method for the inverse scattering problem of a time-harmonic plane wave incident on a perfectly reflecting random periodic structure. The method is based on a novel combination of the Monte Carlo technique for sampling the probability space, a continuation method with respect

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2915-2933, January 2020. The embedded discontinuous Galerkin (EDG) finite element method for the Stokes problem results in a pointwise divergence-free approximate velocity on cells. However, the approximate velocity is not $H({div})$-conforming, and it can be shown that this is the reason that the EDG method is not pressure-robust, i.e.,

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2885-2914, January 2020. In this paper we consider the Runge--Kutta discontinuous Galerkin (RKDG) method to solve linear constant-coefficient hyperbolic equations, where the fourth-order explicit Runge--Kutta time-marching is used. By the aid of the equivalent evolution representation with temporal differences of stage solutions, we make

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2856-2884, January 2020. In this work, we consider conforming finite element discretizations of arbitrary polynomial degree ${p \ge 1}$ of the Poisson problem. We propose a multilevel a posteriori estimator of the algebraic error. We prove that this estimator is reliable and efficient (represents a two-sided bound of the error), with a constant

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2829-2855, January 2020. We study the finite element approximation of the Kirchhoff plate equation on domains with curved boundaries using the Hellan--Herrmann--Johnson (HHJ) method. We prove optimal convergence on domains with piecewise $C^{k+1}$ boundary for $k \geq 1$ when using a parametric (curved) HHJ space. Computational results are

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2818-2828, January 2020. Both the energy method and the Laplace transform method are frequently used for determining the number of boundary conditions required for a well posed initial boundary value problem. We show that these two distinctly different methods yield the same results. The continuous energy method can be mimicked exactly in

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2799-2817, January 2020. We introduce a class of unconditionally energy-stable, high-order-accurate schemes for gradient flows in a very general setting. The new schemes are a high-order analogue of the minimizing-movements approach for generating a time discrete approximation to a gradient flow by solving a sequence of optimization problems

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2764-2798, January 2020. The Stokes equation posed on surfaces is important in some physical models, but its numerical solution poses several challenges not encountered in the corresponding Euclidean setting. These include the fact that the velocity vector should be tangent to the given surface and the possible presence of degenerate modes

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2736-2763, January 2020. A new time discretization method for strongly nonlinear parabolic systems is constructed by combining the fully explicit two-step backward difference formula and a second-order stabilization of wave type. The proposed method linearizes and decouples a nonlinear parabolic system at every time level, with second-order

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2711-2735, January 2020. Trimming is a common operation in computer aided design and, in its simplest formulation, consists in removing superfluous parts from a geometric entity described via splines (a spline patch). After trimming, the geometric description of the patch remains unchanged, but the underlying mesh is unfitted with the physical

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2684-2710, January 2020. We study a two-point flux approximation finite volume scheme for a cross-diffusion system. The scheme is shown to preserve the key properties of the continuous systems, among which the decay of the entropy. The convergence of the scheme is established thanks to compactness properties based on the discrete entropy-entropy

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2662-2683, January 2020. This work is concerned with the proof of a posteriori error estimates for fully discrete Galerkin approximations of the Allen--Cahn equation in two and three spatial dimensions. The numerical method comprises the backward Euler method combined with conforming finite elements in space. For this method, we prove conditional

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2632-2661, January 2020. This paper presents a primal-dual weak Galerkin finite element method for a class of second order elliptic equations of Fokker--Planck type. The method is based on a variational form where all the derivatives are applied to the test functions so that no regularity is necessary for the exact solution of the model equation

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2609-2631, January 2020. This work is concerned with the development of an adaptive space-time numerical method, based on a rigorous a posteriori error bound, for the semilinear heat equation with a general local Lipschitz reaction term whose solution may blow up in finite time. More specifically, conditional a posteriori error bounds are

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2589-2608, January 2020. We analyze the stochastic solutions to the Wigner equation, which explain the nonlocal oscillatory integral operator $\Theta_V$ with an antisymmetric kernel as the generator of two branches of jump processes. All existing branching random walk solutions are formulated based on the Hahn--Jordan decomposition $\The

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2572-2588, January 2020. A new stabilizer free weak Galerkin (WG) method is introduced and analyzed for the biharmonic equation. Stabilizing/penalty terms are often necessary in the finite element formulations with discontinuous approximations to ensure the stability of the methods. Removal of stabilizers will simplify finite element formulations

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2544-2571, January 2020. In this contribution we analyze the large time behavior of a family of nonlinear finite volume schemes for anisotropic convection-diffusion equations set in a bounded bidimensional domain and endowed with either Dirichlet and/or no-flux boundary conditions. We show that solutions to the two-point flux approximation

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2515-2543, January 2020. We consider one-level additive Schwarz preconditioners for a family of Helmholtz problems with increasing wavenumber $k$. These problems are discretized using the Galerkin method with nodal conforming finite elements of any (fixed) order on meshes with diameter $h = h(k)$, chosen to maintain accuracy as $k$ increases

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2492-2514, January 2020. Variable-order space-time fractional diffusion equations, in which the variation of the fractional orders determined by the fractal dimension of the media via the Hurst index characterizes the structure change of porous materials, provide a competitive means to describe anomalously diffusive transport of particles

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2465-2491, January 2020. An efficient numerical scheme based on the scalar auxiliary variable (SAV) and marker and cell (MAC) scheme is constructed for the Navier--Stokes equations. A particular feature of the scheme is that the nonlinear term is treated explicitly while being unconditionally energy stable. A rigorous error analysis is carried

SIAM Journal on Numerical Analysis, Volume 58, Issue 5, Page 2435-2464, January 2020. In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving an integral fractional Laplacian in $\mathbb{R}^d$, which is built upon two essential components: (i) the Dunford--Taylor formulation of the fractional Laplacian; and (ii) Fourier-like biorthogonal mapped Chebyshev functions (MCFs)

SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2404-2433, January 2020. This paper is dedicated to the improvement of the efficiency of the leapfrog method for linear and semilinear second-order differential equations. In numerous situations the strict CFL condition of the leapfrog method is the main bottleneck that thwarts its performance. Based on Chebyshev polynomials new methods have

SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2376-2403, January 2020. This paper is concerned with a priori error estimates for the local incremental minimization scheme, which is an implicit time discretization method for the approximation of rate-independent systems with nonconvex energies. We first show by means of a counterexample that one cannot expect global convergence of the

SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2351-2375, January 2020. In this work, we propose a general discretization framework for numerical solution of forward backward stochastic differential equations (FBSDEs). The framework covers several existing temporal discretization probabilistic schemes in the literature. The consistency, stability, and convergence analysis for the proposed

SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2334-2350, January 2020. We derive and analyze a boundary element formulation for boundary conditions involving inequalities. In particular, we focus on Signorini contact conditions. The Calderón projector is used for the system matrix, and boundary conditions are weakly imposed using a particular variational boundary operator designed using

SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2315-2333, January 2020. Convergence of Dziuk's fully discrete linearly implicit parametric finite element method for curve shortening flow on the plane still remains open since it was proposed in 1991, though the corresponding semidiscrete method with piecewise linear finite elements was proved to be convergent in 1994, while the error analysis

SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2294-2314, January 2020. In this work, we investigate the two-step backward differentiation formula (BDF2) with nonuniform grids for the Allen--Cahn equation. We show that the nonuniform BDF2 scheme is energy stable under the time-step ratio restriction $r_k:=\tau_k/\tau_{k-1}<(3+\sqrt{17})/2\approx3.561.$ Moreover, by developing a novel

SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2265-2293, January 2020. We present a technique for the approximation of a class of Hilbert space--valued maps which arise within the framework of model order reduction (MOR) for parametric partial differential equations, whose solution map has a meromorphic structure. Our MOR strategy consists in constructing an explicit rational approximation

SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2235-2264, January 2020. For incompressible flow models, the pressure term serves as a Lagrange multiplier to ensure that the incompressibility constraint is satisfied. In engineering applications, the pressure term is necessary for calculating important quantities based on stresses like the lift and drag. For reduced-order models generated

SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2212-2234, January 2020. A semilinear 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 L1-type discretizations of this problem, we employ the method of upper and lower solutions to obtain sharp pointwise-in-time

SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2193-2211, January 2020. We analyze the spectrum of the operator $\Delta^{-1} [\nabla \cdot (K\nabla u)]$, where $\Delta$ denotes the Laplacian and $K=K(x,y)$ is a symmetric tensor. Our main result shows that this spectrum can be derived from the spectral decomposition $K=Q \Lambda Q^T$, where $Q=Q(x,y)$ is an orthogonal matrix and $\Lambda=\Lambda(x

SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2165-2192, January 2020. The paper addresses stability and finite element analysis of the stationary two-phase Stokes problem with a piecewise constant viscosity coefficient experiencing a jump across the interface between two fluid phases. We first prove a priori estimates for the individual terms of the Cauchy stress tensor with stability

SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2144-2164, January 2020. Numerical integration is encountered in all fields of numerical analysis and the engineering sciences. By now, various efficient and accurate quadrature rules are known, for instance, Gauss-type quadrature rules. In many applications, however, it might be impractical---if not even impossible---to obtain data to fit

SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2119-2143, January 2020. We present a stability and convergence theory for the lossy Helmholtz equation and its Galerkin discretization. The boundary conditions are of Robin type. All estimates are explicit with respect to the real and imaginary parts of the complex wavenumber $\zeta\in\mathbb{C}$, $\operatorname{Re}\zeta\geq0$, $\left\vert

SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2093-2118, January 2020. We study a model of crowd motion following a gradient vector field, with possibly additional interaction terms such as attraction/repulsion, and we present a numerical scheme for its solution through a Lagrangian discretization. The density constraint of the resulting particles is enforced by means of a partial optimal

SIAM Journal on Numerical Analysis, Volume 58, Issue 4, Page 2079-2092, January 2020. We consider the application of Runge--Kutta (RK) methods to gradient systems $(d/dt)x = -\nabla V(x)$, where, as in many optimization problems, $V$ is convex and $\nabla V$ (globally) Lipschitz-continuous with Lipschitz constant $L$. Solutions of this system behave contractively, i.e., the Euclidean distance between

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