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

SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 1-31, January 2021. In this paper we develop a staggered discontinuous Galerkin method for the Stokes and Darcy--Forchheimer problems coupled with the Beavers--Joseph--Saffman conditions. The method is defined by imposing staggered continuity for all the variables involved and the interface conditions are enforced by switching the roles of

SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3530-3557, January 2020. An artificial boundary method is developed for solving the one-dimensional advection diffusion equation in the real line. In order to construct a fully discrete fast numerical algorithm with rigorous error analysis, we start with the two-step backward difference formula for time discretization of the advection diffusion

SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3504-3529, January 2020. In this paper we study the relation between convergence rates of spectral regularization methods under Hölder-type source conditions resulting from the theory of ill-posed inverse problems, when the noise level $\delta$ goes to 0, and convergence rates resulting from statistical kernel learning, when the number of

SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3475-3503, January 2020. For implicit-explicit multistep schemes, using a suitable form of Dahlquist's test equation, we introduce an analogue to the A($\vartheta$)-stability concept, valid for implicit methods, and formulate a stability criterion in terms of an auxiliary function that plays a key role in our analysis. Furthermore, for implicit-explicit

SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3448-3474, January 2020. In this article a modification of the algorithm for data discretized in the point values introduced in [S. Amat, J. Ruiz, and C.-W. Shu, Appl. Math. Lett., 105 (2020), pp. 106--298] is presented. In the aforementioned work, the authors managed to obtain an algorithm that reaches a progressive and optimal order of

SIAM Journal on Numerical Analysis, Volume 58, Issue 6, Page 3427-3447, January 2020. Partial outer convexification has been used to derive relaxations of mixed-integer optimal control problems (MIOCPs) that are constrained by time-dependent differential equations. The family of sum-up rounding (SUR) algorithms provides a means to approximate feasible points of these relaxations, i.e., $[0,1]$-valued

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 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 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