SIAM Journal on Numerical Analysis, Volume 59, Issue 5, Page 2368-2392, January 2021. We design and analyze an arbitrary-order hybridized discontinuous Galerkin method to approximate the unique continuation problem subject to the Helmholtz equation. The method is analyzed using conditional stability estimates for the continuous problem, leading to error estimates in norms over interior subdomains of

SIAM Journal on Numerical Analysis, Volume 59, Issue 5, Page 2346-2367, January 2021. In this paper we propose a method for computing the Faddeeva function $w(z) := {e}^{-z^2}{erfc}(-{i}\,z)$ via truncated modified trapezoidal rule approximations to integrals on the real line. Our starting point is the method due to Matta and Reichel [Math. Comp., 25 (1971), pp. 339--344] and Hunter and Regan [Math

SIAM Journal on Numerical Analysis, Volume 59, Issue 5, Page 2321-2345, January 2021. We develop a new finite difference scheme for the Maxwell--Stefan diffusion system. The scheme is conservative, energy-stable, and positivity-preserving. These nice properties stem from a variational structure and are proved by reformulating the finite difference scheme into an equivalent optimization problem. The

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2310-2319, January 2021. We present an algorithm for the efficient numerical evaluation of integrals of the form $I(\omega) = \int_0^1 F( x,\e^{\i \omega x}; \omega) \, \d x$ for sufficiently smooth but otherwise arbitrary $F$ and $\omega \gg 1$. The method is entirely “black-box,” i.e., it does not require the explicit computation of moment

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2286-2309, January 2021. An implicit Euler finite-volume scheme for an $n$-species population cross-diffusion system of Shigesada--Kawasaki--Teramoto-type in a bounded domain with no-flux boundary conditions is proposed and analyzed. The scheme preserves the formal gradient-flow or entropy structure and preserves the nonnegativity of the

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2254-2285, January 2021. Dynamical low-rank algorithms are a class of numerical methods that compute low-rank approximations of dynamical systems. This is accomplished by projecting the dynamics onto a low-dimensional manifold and writing the solution directly in terms of the low-rank factors. The approach has been successfully applied to

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2237-2253, January 2021. We present an equilibration-based a posteriori error estimator for Nédélec element discretizations of the magnetostatic problem. The estimator is obtained by adding a gradient correction to the estimator for Nédélec elements of arbitrary degree presented in [Gedicke, Geevers, and Perugia, J. Sci. Comput., 83 (2020)

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2218-2236, January 2021. We consider systems of stochastic evolutionary equations of p-Laplace type. We establish convergence rates for a finite-element-based space-time approximation, where the error is measured in a suitable quasi-norm. Under natural regularity assumptions on the solution, our main result provides linear convergence in

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2197-2217, January 2021. This paper proposes a fully discrete method called the symplectic discontinuous Galerkin (dG) full discretization for stochastic Maxwell equations driven by additive noises, based on a stochastic symplectic method in time and a dG method with the upwind fluxes in space. A priori $H^k$-regularity ($k\in\{1,2\}$) estimates

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2163-2196, January 2021. In this paper, we resolve several long-standing issues dealing with optimal pointwise in time error bounds for proper orthogonal decomposition (POD) reduced order modeling of the heat equation. In particular, we study the role played by difference quotients (DQs) in obtaining reduced order model (ROM) error bounds

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2138-2162, January 2021. The numerical integration over a planar domain that is cut by an implicitly defined boundary curve is an important problem that arises, for example, in unfitted finite element methods and in isogeometric analysis on trimmed computational domains. In this paper, we introduce a a very general version of the transport

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2106-2137, January 2021. This paper proposes a simple and rigorous Fourier-matching method to study transverse-magnetic-polarized electro-magnetic resonances by a perfectly conducting slab with a finite number of subwavelength slits of width $h\ll 1$. Since variable separation is applicable in the region outside the slits, by Fourier transforming

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2075-2105, January 2021. In this work, we consider the error estimates of some splitting schemes for the charged-particle dynamics under a strong magnetic field. We first propose a novel energy-preserving splitting scheme with computational cost per step independent of the strength of the magnetic field. Then under the maximal ordering scaling

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2054-2074, January 2021. We consider radial complex scaling/perfectly matched layer methods for scalar resonance problems in homogeneous exterior domains. We introduce a new abstract framework to analyze the convergence of domain truncations and discretizations. Our theory requires rather minimal assumptions on the scaling profile and includes

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2040-2053, January 2021. We present general criteria ensuring the positivity of quadratic forms of convolution type generated by sequences of real numbers. A sharp result is obtained in the case of completely monotone sequences. Applications to widely used approximations of fractional integral and differential operators, including convolution

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2004-2039, January 2021. This work is dedicated to the proof of stability and convergence of the Bérenger's perfectly matched layers (PMLs) in the waveguides for an arbitrary nonnegative $L^{\infty}$ damping function. The proof relies on the Laplace-domain techniques and an explicit representation of the solution to the PML problem in the

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 1976-2003, January 2021. The full discretization of the semilinear stochastic wave equation is considered. The discontinuous Galerkin finite element method is used in space and analyzed in a semigroup framework, and an explicit stochastic position Verlet scheme is used for the temporal approximation. We study the stability under a CFL condition

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 1948-1975, January 2021. The estimation of the probability of rare events is an important task in reliability and risk assessment. We consider failure events that are expressed in terms of a limit-state function, which depends on the solution of a partial differential equation (PDE). In many applications, the PDE cannot be solved analytically

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 1918-1947, January 2021. The integral fractional Laplacian of order $s \in (0,1)$ is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singularity regardless of the domain regularity. This, in turn, deteriorates the global regularity of solutions and as a result

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 1896-1917, January 2021. We describe a new approximation method for solving a PDE-constrained optimization problem numerically. Our method is based on the adjoint formulation of the optimization problem, leading to a system of weakly coupled, elliptic PDEs. These equations are then solved using kernel-based collocation. We derive an error

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 1875-1895, January 2021. The time discrete scheme of characteristics type is especially effective for convection-dominated diffusion problems. The scheme has been used in various engineering areas with different approximations in spatial direction. The lowest-order mixed method is the most popular one for miscible flow in porous media. The

SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 1857-1874, January 2021. Pseudospectral (PS) methods are widely used for solving partial differential equations (PDEs) to high accuracy in simple geometries. They can be seen as the limit of finite difference methods for increasing order of accuracy. Similarly, recently introduced Hermite-based finite difference approximations converge to

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1835-1856, January 2021. Piecewise divergence-free nonconforming virtual elements are designed for Stokes problem in any dimensions. After introducing a local energy projector based on the Stokes problem and the stabilization, a divergence-free nonconforming virtual element method is proposed for Stokes problem. A detailed and rigorous error

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1811-1834, January 2021. We propose a supervised deep learning algorithm based on low-discrepancy sequences as the training set. By a combination of theoretical arguments and extensive numerical experiments we demonstrate that the proposed algorithm significantly outperforms standard deep learning algorithms that are based on randomly chosen

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1769-1810, January 2021. Homographic complex best approximation has emerged in recent years as an essential tool for the design of new, performant domain decomposition Robin--Schwarz algorithms. We present and analyze a fully complex problem, introducing a new alternation property. We give operational formulas for the solution and apply them

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1735-1768, January 2021. We introduce a new general framework for the approximation of evolution equations at low regularity and develop a new class of schemes for a wide range of equations under lower regularity assumptions than classical methods require. In contrast to previous works, our new framework allows a unified practical formulation

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1687-1734, January 2021. Diffusion maps is a manifold learning algorithm widely used for dimensionality reduction. Using a sample from a distribution, it approximates the eigenvalues and eigenfunctions of associated Laplace--Beltrami operators. Theoretical bounds on the approximation error are, however, generally much weaker than the rates

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1663-1686, January 2021. Generalized plane waves (GPWs) were introduced to take advantage of Trefftz methods for problems modeled by variable coefficient equations. Despite the fact that GPWs do not satisfy the Trefftz property, i.e., they are not exact solutions to the governing equation, they instead satisfy a quasi-Trefftz property: They

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1639-1662, January 2021. The Landau--Lifshitz equation has been widely used to describe the dynamics of magnetization in a ferromagnetic material, which is highly nonlinear with the nonconvex constraint $|{m}|=1$. A crucial issue in designing efficient numerical schemes is to preserve this constraint in the discrete level. A simple and frequently

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1618-1638, January 2021. Explicit time advancement for continuous finite elements requires the inversion of a global mass matrix. For spectral element simulations on quadrilaterals and hexahedra, there is an accurate approximate mass matrix which is diagonal, making it computationally efficient for explicit simulations. In this article it

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1592-1617, January 2021. Dziuk's surface finite element method (FEM) for mean curvature flow has had a significant impact on the development of parametric and evolving surface FEMs for surface evolution equations and curvature flows. However, the convergence of Dziuk's surface FEM for mean curvature flow of closed surfaces still remains open

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1566-1591, January 2021. A family of arbitrarily high-order fully discrete space-time finite element methods are proposed for the nonlinear Schrödinger equation based on the scalar auxiliary variable formulation, which consists of a Gauss collocation temporal discretization and the finite element spatial discretization. The proposed methods

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1542-1565, January 2021. In this manuscript we study the properties of a family of a second-order differential equations with damping, its discretizations, and their connections with accelerated optimization algorithms for $m$-strongly convex and $L$-smooth functions. In particular, using the linear matrix inequality (LMI) framework developed

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1510-1541, January 2021. Shape gradients have been widely used in numerical shape gradient descent algorithms for shape optimization. The two types of shape gradients, i.e., the distributed one and the boundary type, are equivalent at the continuous level but exhibit different numerical behaviors after finite element discretization. To be

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1486-1509, January 2021. We deal with the numerical solution of nonlinear time-dependent convection-diffusion-reaction equations with the aid of continuous and discontinuous Galerkin discretization of an arbitrary polynomial approximation degree. We derive a posteriori error estimates in the space-time mesh-dependent dual norm. The estimates

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1455-1485, January 2021. We propose two algorithms for the solution of the optimal control of ergodic McKean--Vlasov dynamics. Both algorithms are based on approximations of the theoretical solutions by neural networks, the latter being characterized by their architecture and a set of parameters. This allows the use of modern machine learning

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1433-1454, January 2021. Optimal order error estimates of the energy-preserving numerical methods for solving the Hodge wave equation is obtained in this paper. Based on the de Rham complex, the Hodge wave equation can be formulated as a first-order system and mixed finite element methods using finite element exterior calculus is used to

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1399-1432, January 2021. This paper presents a novel mathematical framework for understanding pixel-driven approaches for the parallel beam Radon transform as well as for the fanbeam transform, showing that with the correct discretization strategy, convergence---including rates---in the $L^2$ operator norm can be obtained. From these rates

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1374-1398, January 2021. Based on the Hilb type formula between Jacobi polynomials and Bessel functions, optimal decay rates on the Jacobi expansion coefficients are derived by applying van der Corput type lemmas for functions of algebraic and logarithmatic singularities, which leads to the optimal convergence rates on the Jacobi, Gegenbauer

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1348-1373, January 2021. The relaxation in the calculus of variation motivates the numerical analysis of a class of degenerate convex minimization problems with nonstrictly convex energy densities with some convexity control and two-sided $p$-growth. The minimizers may be nonunique in the primal variable but lead to a unique stress $\sigma

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1325-1347, January 2021. This work develops novel rational Krylov methods for updating a large-scale matrix function $f(A)$ when $A$ is subject to low-rank modifications. It extends our previous work in this context on polynomial Krylov methods, for which we present a simplified convergence analysis. For the rational case, our convergence

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1299-1324, January 2021. In this paper, we propose a novel discontinuous Galerkin (DG) method to control the spurious oscillations when solving the scalar hyperbolic conservation laws. Usually, the high order linear numerical schemes would generate spurious oscillations when the solution of the hyperbolic conservation laws contains discontinuities

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1273-1298, January 2021. We introduce a residual-based a posteriori error estimator for a novel $hp$-version interior penalty discontinuous Galerkin method for the biharmonic problem in two and three dimensions. We prove that the error estimate provides an upper bound and a local lower bound on the error, and that the lower bound is robust

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1245-1272, January 2021. This paper is the second part in a series on a mass conserving, high order, mixed finite element method for Stokes flow. In this part, we construct optimal $hp$-approximations for two classes of functions: (1) functions with general Sobolev regularity which lead to algebraic convergence rates using locally quasi-uniform

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1218-1244, January 2021. We study the conforming finite element spaces $\Sigma_{0}^{h,k}(\Omega) \subset H^2_0(\Omega)$, ${V}^{h,k}_0(\Omega) \subset {H}^1_0(\Omega)$, and $Q_{0}^{h,k}(\Omega) \subset L^2_0(\Omega)$ recently proposed by Falk and Neilan [SIAM J. Numer. Anal., 51 (2013), pp. 1308--1326] and establish necessary and sufficient

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1195-1217, January 2021. A shear-locking free finite element discretization of the Reissner--Mindlin plate model is introduced. The rotation is discretized with piecewise polynomials of degree $k+2$ while the degree $k\geq0$ is used for the displacement gradient. The method is closely related to the (generalized) Taylor--Hood pairing. In

SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1167-1194, January 2021. We present a second-order accurate numerical method for a class of nonlocal nonlinear conservation laws called the “nonlocal pair-interaction model,” which was recently introduced by Du, Huang, and LeFloch [SIAM J. Numer. Anal., 55 (2017), pp. 2465--2489]. Our numerical method uses second-order accurate reconstruction-based

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 1140-1165, January 2021. We consider the integral definition of the fractional Laplacian and analyze a linear-quadratic optimal control problem for the so-called fractional heat equation; control constraints are also considered. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 1117-1139, January 2021. In this work we consider the dual-mixed variational formulation of the Poisson equation with a line source. The analysis and approximation of this problem is nonstandard, as the line source causes the solutions to be singular. We start by showing that this problem admits a solution in appropriately weighted Sobolev

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 1090-1116, January 2021. We construct and analyze an isoparametric finite element pair for the Stokes problem in two dimensions. The pair is defined by mapping the Scott--Vogelius finite element space via a Piola transform. The velocity space has the same degrees of freedom as the quadratic Lagrange finite element space, and therefore the

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 1067-1089, January 2021. We propose a multiscale approach for an elliptic multiscale setting with general unstructured diffusion coefficients that is able to achieve high-order convergence rates with respect to the mesh parameter and the polynomial degree. The method allows for suitable localization and does not rely on additional regularity

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 1040-1066, January 2021. Superresolution of the Lie--Trotter splitting ($S_1$) and Strang splitting ($S_2$) is rigorously analyzed for the nonlinear Dirac equation without external magnetic potentials in the nonrelativistic regime with a small parameter $0<{\varepsilon}\leq 1$ inversely proportional to the speed of light. In this regime,

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 998-1039, January 2021. We present a globally convergent numerical algorithm based on global Carleman estimates to reconstruct the speed of wave propagation in a bounded domain with Dirichlet boundary conditions from a single measurement of the boundary flux of the solutions in a finite time interval. The global convergence of the proposed

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 983-997, January 2021. A proof for the lower bound is provided for the smallest eigenvalue of finite element equations with arbitrary conforming simplicial meshes. The bound has a similar form to the one by Graham and McLean [SIAM J. Numer. Anal., 44 (2006), pp. 1487--1513] but doesn't require any mesh regularity assumptions, neither global

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 955-982, January 2021. This paper establishes the convergence of adaptive mixed finite element methods for second-order linear non-self-adjoint indefinite elliptic problems in three dimensions with piecewise Lipschitz continuous coefficients. The error is measured in the $L^2$ norms and then allows for an adaptive algorithm with collective

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 925-954, January 2021. In our recent work [Z. Peng et al., J. Comput. Phys., 415 (2020), 109485], a family of high-order asymptotic preserving (AP) methods, termed IMEX-LDG methods, are designed to solve some linear kinetic transport equations, including the one-group transport equation in slab geometry and the telegraph equation, in a diffusive

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 886-924, January 2021. This paper presents a new narrow-stencil finite difference method for approximating viscosity solutions of Hamilton--Jacobi--Bellman equations. The proposed finite difference scheme naturally extends the Lax--Friedrichs scheme for first order fully nonlinear PDEs to second order fully nonlinear PDEs which are approximated

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 864-885, January 2021. We consider the numerical approximation of Maxwell's equations in the time domain by a second-order $H({curl})$ conforming finite element approximation. In order to enable the efficient application of explicit time-stepping schemes, we utilize a mass-lumping strategy resulting from numerical integration in conjunction

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 829-863, January 2021. We consider a finite element method for elliptic equations with heterogeneous and possibly high-contrast coefficients based on primal hybrid formulation. We assume minimal regularity of the solutions. A space decomposition as in FETI and BDCC induces an embarrassingly parallel preprocessing and leads to a final system

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 797-828, January 2021. Immersed finite element (IFE) methods are a group of long-existing numerical methods for solving interface problems on unfitted meshes. A core argument of the methods is to avoid a mesh regeneration procedure when solving moving interface problems. Despite the various applications in moving interface problems, a complete