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

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 769-796, January 2021. We study the Schwarz overlapping domain decomposition method applied to the Poisson problem on a special family of domains, which by construction consist of a union of a large number of fixed-size subdomains. These domains are motivated by applications in computational chemistry where the subdomains consist of van der

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 746-768, January 2021. We consider in this work the convergence of the random batch method proposed in our previous work [Jin et al., J. Comput. Phys., 400(2020), 108877] for interacting particles to the case of disparate species and weights. We show that the strong error is of $O(\sqrt{\tau})$ while the weak error is of $O(\tau)$ where $\tau$

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 720-745, January 2021. We prove quasi-optimal $L^\infty$ norm error estimates (up to logarithmic factors) for the solution of Poisson's problem in two dimensional space by the standard hybridizable discontinuous Galerkin (HDG) method. Although such estimates are available for conforming and mixed finite element methods, this is the first

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 696-719, January 2021. The adaptive nonconforming Morley finite element method approximates a regular solution to the von Kármán equations with optimal convergence rates for sufficiently fine triangulations and small bulk parameter in the Dörfler marking. This follows from the general axiomatic framework with the key arguments of stability

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 675-695, January 2021. In this paper, we analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{-1}(\Omega))$. By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 660-674, January 2021. This paper provides an a priori error analysis of a localized orthogonal decomposition method for the numerical stochastic homogenization of a model random diffusion problem. If the uniformly elliptic and bounded random coefficient field of the model problem is stationary and satisfies a quantitative decorrelation assumption

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 634-659, January 2021. Problems in astrophysics, space weather research, and geophysics usually need to analyze big noisy data on the sphere. This paper develops distributed filtered hyperinterpolation for noisy data on the sphere, which assigns the data fitting task to multiple servers to find a good approximation of the mapping of input

SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 613-633, January 2021. Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier--Galerkin spectral method [L. Pareschi and G. Russo, SIAM J. Numer. Anal., 37 (2000), pp. 1217--1245] has become a popular deterministic

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 583-612, January 2021. In this work, we consider compressible single-phase flow problems in a porous medium containing a fracture. In the fracture, a nonlinear pressure-velocity relation is prescribed. Using a non-overlapping domain decomposition procedure, we reformulate the global problem into a nonlinear interface problem. We then introduce

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 558-582, January 2021. Coupled partial differential equations (PDEs) defined on domains with different dimensionality are usually called mixed-dimensional PDEs. We address mixed-dimensional PDEs on three-dimensional (3D) and one-dimensional (1D) domains, which gives rise to a 3D-1D coupled problem. Such a problem poses several challenges

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 525-557, January 2021. This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming. Unlike classical approximate dynamic programming approaches, we first approximate the optimal policy by means of neural networks in the spirit of deep reinforcement learning, and then the value

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 503-524, January 2021. We introduce a new discretization for the stream function formulation of the incompressible Stokes equations in two and three space dimensions. The method is strongly related to the Hellan--Herrmann--Johnson method and is based on the recently discovered mass conserving mixed stress formulation [J. Gopalakrishnan, P

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 477-502, January 2021. We study the rate of weak convergence of Markov chains to diffusion processes under quite general assumptions. We give an example in the financial framework, applying the convergence analysis to a multiple jumps tree approximation of the CIR process. Then, we combine the Markov chain approach with other numerical techniques

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 456-476, January 2021. Boundary perturbation methods have received considerable attention in recent years due to their ability to simulate solutions of differential equations of applied interest in a stable, robust, and highly accurate fashion. In this contribution we study the rigorous numerical analysis of a recently proposed high-order

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 429-455, January 2021. Linear multistep methods (LMMs) are popular time discretization techniques for the numerical solution of differential equations. Traditionally they are applied to solve for the state given the dynamics (the forward problem), but here we consider their application for learning the dynamics given the state (the inverse

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 401-428, January 2021. We propose and analyze an efficient, unconditionally stable, second order convergent, artificial compressibility Crank--Nicolson leap-frog (CNLFAC) method for numerically solving the Stokes--Darcy equations. The method decouples the fully coupled Stokes--Darcy system into two smaller subphysics problems, which reduces

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 370-400, January 2021. We study a stability preserved Arnoldi algorithm for matrix exponential in the time domain simulation of large-scale power delivery networks (PDNs), which are formulated as semi-explicit differential algebraic equations (DAEs). The solution can be decomposed to a sum of two projections, one in the range of the system

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 334-369, January 2021. Proper orthogonal decomposition (POD) stabilized methods for the Navier--Stokes equations are considered and analyzed. We consider two cases: the case in which the snapshots are based on a non inf-sup stable method and the case in which the snapshots are based on an inf-sup stable method. For both cases we construct

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 314-333, January 2021. An important observation in compressed sensing is that the $\ell_0$ minimizer of an underdetermined linear system is equal to the $\ell_1$ minimizer when there exists a sparse solution vector and a certain restricted isometry property holds. Here, we develop a continuous analogue of this observation and show that the

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 289-313, January 2021. Dynamical low-rank approximation by tree tensor networks is studied for the data-sparse approximation of large time-dependent data tensors and unknown solutions to tensor differential equations. A time integration method for tree tensor networks of prescribed tree rank is presented and analyzed. It extends the known

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 265-288, January 2021. An implicit energy-decaying modified Crank--Nicolson time-stepping method is constructed for the viscous rotating shallow water equation on the plane. Existence, uniqueness, and convergence of semidiscrete solutions are proved by using Schaefer's fixed point theorem and $H^2$ estimates of the discretized hyperbolic--parabolic

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 249-264, January 2021. Two a posteriori error estimates for a numerical approximation scheme for the inf-sup constant for the divergence (also known as the LBB constant) are shown. Under the assumption that the inf-sup constant is an eigenvalue of the Cosserat operator separated from the essential spectrum and that the mesh size is sufficiently

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 219-248, January 2021. The Cahn--Hilliard equation is a widely used model that describes among others phase-separation processes of binary mixtures or two-phase flows. In recent years, different types of boundary conditions for the Cahn--Hilliard equation were proposed and analyzed. In this publication, we are concerned with the numerical

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 195-218, January 2021. This paper deals with the derivation and analysis of reduced order elliptic PDE models on fractured domains. We use a Fourier analysis to obtain coupling conditions between subdomains when the fracture is represented as a hypersurface embedded in the surrounded rock matrix. We compare our results to prominent examples

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 173-194, January 2021. This work is concerned with the development of a family of Galerkin finite element methods for the classical Kolmogorov equation. Kolmogorov's equation serves as a sufficiently rich, for our purposes, model problem for kinetic-type equations and is characterized by diffusion in one of the two (or three) spatial directions

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 143-172, January 2021. The aim of this study is to develop a novel WENO scheme that improves the performance of the well-known fifth-order WENO methods. The approximation space consists of exponential polynomials with a tension parameter that may be optimized to fit the the specific feature of the data, yielding better results compared to

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 119-142, January 2021. In this work, we present a novel error analysis for recovering a spatially dependent diffusion coefficient in an elliptic or parabolic problem. It is based on the standard regularized output least-squares formulation with an $H^1(\Omega)$ seminorm penalty and then discretized using the Galerkin finite element method

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 88-118, January 2021. Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation method for nonlocal diffusion models with Dirichlet boundary condition. The numerical scheme is shown to be convergent to both nonlocal diffusion

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 60-87, January 2021. Various degenerate diffusion equations exhibit a waiting time phenomenon: depending on the “flatness” of the compactly supported initial datum at the boundary of the support, the support of the solution may not expand for a certain amount of time. We show that this phenomenon is captured by particular Lagrangian discretizations

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 32-59, January 2021. It is well known that symplectic methods have been rigorously shown to be superior to nonsymplectic ones especially in long-time computation, when applied to deterministic Hamiltonian systems. In this paper, we attempt to study the superiority of stochastic symplectic methods by means of the large deviations principle