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Optimal convergence analysis of two RPC-SAV schemes for the unsteady incompressible magnetohydrodynamics equations IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-05-15 Xiaojing Dong, Huayi Huang, Yunqing Huang, Xiaojuan Shen, Qili Tang
In this paper, we present and analyze two linear and fully decoupled schemes for solving the unsteady incompressible magnetohydrodynamics equations. The rotational pressure-correction (RPC) approach is adopted to decouple the system, and the recently developed scalar auxiliary variable (SAV) method is used to treat the nonlinear terms explicitly and keep energy stability. One is the first-order RPC-SAV-Euler
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Efficient function approximation in enriched approximation spaces IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-05-12 Astrid Herremans, Daan Huybrechs
An enriched approximation space is the span of a conventional basis with a few extra functions included, for example to capture known features of the solution to a computational problem. Adding functions to a basis makes it overcomplete and, consequently, the corresponding discretized approximation problem may require solving an ill-conditioned system. Recent research indicates that these systems can
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Error analysis for local discontinuous Galerkin semidiscretization of Richards’ equation IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-05-11 Scott Congreve, Vít Dolejší, Sunčica Sakić
This paper concerns an error analysis of the space semidiscrete scheme for the Richards’ equation modeling flows in variably saturated porous media. This nonlinear parabolic partial differential equation can degenerate; namely, we consider the case where the time derivative term can vanish, i.e., the fast-diffusion type of degeneracy. We discretize the Richards’ equation by the local discontinuous
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Variational data assimilation with finite-element discretization for second-order parabolic interface equation IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-05-11 Xuejian Li, Xiaoming He, Wei Gong, Craig C Douglas
In this paper, we propose and analyze a finite-element method of variational data assimilation for a second-order parabolic interface equation on a two-dimensional bounded domain. The Tikhonov regularization plays a key role in translating the data assimilation problem into an optimization problem. Then the existence, uniqueness and stability are analyzed for the solution of the optimization problem
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A linearly implicit finite element full-discretization scheme for SPDEs with nonglobally Lipschitz coefficients IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-05-08 Mengchao Wang, Xiaojie Wang
The present article deals with strong approximations of additive noise driven stochastic partial differential equations (SPDEs) with nonglobally Lipschitz nonlinearity in a bounded domain $ \mathcal{D} \in{\mathbb{R}}^{d}$, $ d \leq 3$. As the first contribution, we establish the well-posedness and regularity of the considered SPDEs in space dimension $d \le 3$, under more relaxed assumptions on the
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Stability of convergence rates: kernel interpolation on non-Lipschitz domains IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-05-08 Tizian Wenzel, Gabriele Santin, Bernard Haasdonk
Error estimates for kernel interpolation in Reproducing Kernel Hilbert Spaces usually assume quite restrictive properties on the shape of the domain, especially in the case of infinitely smooth kernels like the popular Gaussian kernel. In this paper we prove that it is possible to obtain convergence results (in the number of interpolation points) for kernel interpolation for arbitrary domains $\varOmega
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A certified wavelet-based physics-informed neural network for the solution of parameterized partial differential equations IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-05-05 Lewin Ernst, Karsten Urban
Physics Informed Neural Networks (PINNs) have frequently been used for the numerical approximation of Partial Differential Equations (PDEs). The goal of this paper is to construct PINNs along with a computable upper bound of the error, which is particularly relevant for model reduction of Parameterized PDEs (PPDEs). To this end, we suggest to use a weighted sum of expansion coefficients of the residual
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Strong convergence of adaptive time-stepping schemes for the stochastic Allen–Cahn equation IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-05-05 Chuchu Chen, Tonghe Dang, Jialin Hong
It is known from Beccari et al. (2019) that the standard explicit Euler-type scheme (such as the exponential Euler and the linear-implicit Euler schemes) with a uniform timestep, though computationally efficient, may diverge for the stochastic Allen–Cahn equation. To overcome the divergence, this paper proposes and analyzes adaptive time-stepping schemes, which adapt the timestep at each iteration
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Finite element methods for multicomponent convection-diffusion IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-04-28 Francis R A Aznaran, Patrick E Farrell, Charles W Monroe, Alexander J Van-Brunt
We develop finite element methods for coupling the steady-state Onsager–Stefan–Maxwell (OSM) equations to compressible Stokes flow. These equations describe multicomponent flow at low Reynolds number, where a mixture of different chemical species within a common thermodynamic phase is transported by convection and molecular diffusion. Developing a variational formulation for discretizing these equations
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An explicit spectral Fletcher–Reeves conjugate gradient method for bi-criteria optimization IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-04-12 Y Elboulqe, M El Maghri
In this paper, we propose a spectral Fletcher–Reeves conjugate gradient-like method for solving unconstrained bi-criteria minimization problems without using any technique of scalarization. We suggest an explicit formulae for computing a descent direction common to both criteria. The latter further verifies a sufficient descent property that does not depend on the line search nor on any convexity assumption
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On the rate of convergence of Yosida approximation for the nonlocal Cahn–Hilliard equation IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-04-10 Piotr Gwiazda, Jakub Skrzeczkowski, Lara Trussardi
It is well-known that one can construct solutions to the nonlocal Cahn–Hilliard equation with singular potentials via Yosida approximation with parameter $\lambda \to 0$. The usual method is based on compactness arguments and does not provide any rate of convergence. Here, we fill the gap and we obtain an explicit convergence rate $\sqrt{\lambda }$. The proof is based on the theory of maximal monotone
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Monolithic and local time-stepping decoupled algorithms for transport problems in fractured porous media IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-04-03 Yanzhao Cao, Thi-Thao-Phuong Hoang, Phuoc-Toan Huynh
The objective of this paper is to develop efficient numerical algorithms for the linear advection-diffusion equation in fractured porous media. A reduced fracture model is considered where the fractures are treated as interfaces between subdomains and the interactions between the fractures and the surrounding porous medium are taken into account. The model is discretized by a backward Euler upwind-mixed
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On the necessity of the inf-sup condition for a mixed finite element formulation IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-02-28 Fleurianne Bertrand, Daniele Boffi
We study a nonstandard mixed formulation of the Poisson problem, sometimes known as dual mixed formulation. For reasons related to the equilibration of the flux, we use finite elements that are conforming in $\textbf{H}(\operatorname{\textrm{div}};\varOmega )$ for the approximation of the gradients, even if the formulation would allow for discontinuous finite elements. The scheme is not uniformly inf-sup
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Pressure and convection robust bounds for continuous interior penalty divergence-free finite element methods for the incompressible Navier–Stokes equations IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-02-07 Bosco García-Archilla, Julia Novo
In this paper, we analyze a pressure-robust method based on divergence-free mixed finite element methods with continuous interior penalty stabilization. The main goal is to prove an $O(h^{k+1/2})$ error estimate for the $L^2$ norm of the velocity in the convection dominated regime. This bound is pressure robust (the error bound of the velocity does not depend on the pressure) and also convection robust
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An Eulerian finite element method for the linearized Navier–Stokes problem in an evolving domain IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-01-30 Michael Neilan, Maxim Olshanskii
The paper addresses an error analysis of an Eulerian finite element method used for solving a linearized Navier–Stokes problem in a time-dependent domain. In this study, the domain’s evolution is assumed to be known and independent of the solution to the problem at hand. The numerical method employed in the study combines a standard backward differentiation formula-type time-stepping procedure with
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Goal-oriented error estimation based on equilibrated flux reconstruction for the approximation of the harmonic formulations in eddy current problems IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-01-29 Emmanuel Creusé, Serge Nicaise, Zuqi Tang
In this work, we propose an a posteriori goal-oriented error estimator for the harmonic $\textbf {A}$-$\varphi $ formulation arising in the modeling of eddy current problems, approximated by nonconforming finite element methods. It is based on the resolution of an adjoint problem associated with the initial one. For each of these two problems, a guaranteed equilibrated estimator is developed using
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Cauchy data for Levin’s method IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-01-25 Anthony Ashton
In this paper, we describe the Cauchy data that gives rise to slowly oscillating solutions to the Levin equation. We present a general result on the existence of a unique minimizer of $\|Bx\|$ subject to the constraint $Ax=y$, where $A,B$ are linear, but not necessarily bounded operators on a complex Hilbert space. This result is used to obtain the solution to the Levin equation, both in the univariate
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Full operator preconditioning and the accuracy of solving linear systems IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-01-25 Stephan Mohr, Yuji Nakatsukasa, Carolina Urzúa-Torres
Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error and often slow convergence of an iterative solver. In many cases, such systems arise from the discretization of operator equations with a large number of discrete variables and the ill-conditioning is tackled by means of preconditioning
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Well-posedness and error estimates for coupled systems of nonlocal conservation laws IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-01-20 Aekta Aggarwal, Helge Holden, Ganesh Vaidya
This article deals with the error estimates for numerical approximations of the entropy solutions of coupled systems of nonlocal hyperbolic conservation laws. The systems can be strongly coupled through the nonlocal coefficient present in the convection term. A fairly general class of fluxes is being considered, where the local part of the flux can be discontinuous at infinitely many points, with possible
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Corrigendum to: Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-01-19 Maximilian Brunner, Michael Innerberger, Ani Miraçi, Dirk Praetorius, Julian Streitberger, Pascal Heid
Unfortunately, there is a flaw in the numerical analysis of the published version [IMA J. Numer. Anal., DOI:10.1093/imanum/drad039], which is corrected here. Neither the algorithm nor the results are affected, but constants have to be adjusted.
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Convergence with rates for a Riccati-based discretization of SLQ problems with SPDEs IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-01-19 Andreas Prohl, Yanqing Wang
We consider a new discretization in space (parameter $h>0$) and time (parameter $\tau>0$) of a stochastic optimal control problem, where a quadratic functional is minimized subject to a linear stochastic heat equation with linear noise. Its construction is based on the perturbation of a generalized difference Riccati equation to approximate the related feedback law. We prove a convergence rate of almost
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Gamma-convergent LDG method for large bending deformations of bilayer plates IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-01-19 Andrea Bonito, Ricardo H Nochetto, Shuo Yang
Bilayer plates are slender structures made of two thin layers of different materials. They react to environmental stimuli and undergo large bending deformations with relatively small actuation. The reduced model is a constrained minimization problem for the second fundamental form, with a given spontaneous curvature that encodes material properties, subject to an isometry constraint. We design a local
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Darcy’s problem coupled with the heat equation under singular forcing: analysis and discretization IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-01-11 Alejandro Allendes, Gilberto Campaña, Francisco Fuica, Enrique Otárola
We study the existence of solutions for Darcy’s problem coupled with the heat equation under singular forcing; the right-hand side of the heat equation corresponds to a Dirac measure. The model studied involves thermal diffusion and viscosity depending on the temperature. We propose a finite element solution technique and analyze its convergence properties. In the case where thermal diffusion is independent
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Compactness estimates for difference schemes for conservation laws with discontinuous flux IMA J. Numer. Anal. (IF 2.1) Pub Date : 2024-01-04 Kenneth H Karlsen, John D Towers
We establish quantitative compactness estimates for finite difference schemes used to solve nonlinear conservation laws. These equations involve a flux function $f(k(x,t),u)$, where the coefficient $k(x,t)$ is $BV$-regular and may exhibit discontinuities along curves in the $(x,t)$ plane. Our approach, which is technically elementary, relies on a discrete interaction estimate and one entropy function
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A constraint dissolving approach for nonsmooth optimization over the Stiefel manifold IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-12-23 Xiaoyin Hu, Nachuan Xiao, Xin Liu, Kim-Chuan Toh
This paper focuses on the minimization of a possibly nonsmooth objective function over the Stiefel manifold. The existing approaches either lack efficiency or can only tackle prox-friendly objective functions. We propose a constraint dissolving function named NCDF and show that it has the same first-order stationary points and local minimizers as the original problem in a neighborhood of the Stiefel
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On best p-norm approximation of discrete data by polynomials IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-12-09 Michael S Floater
In this note, we derive a solution to the problem of finding a polynomial of degree at most $n$ that best approximates data at $n+2$ points in the $l_{p}$ norm. Analogous to a result of de la Vallée Poussin, one can express the solution as a convex combination of the Lagrange interpolants over subsets of $n+1$ points, and the error oscillates in sign.
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Semilinear optimal control with Dirac measures IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-11-29 Enrique Otárola
The purpose of this work is to study an optimal control problem for a semilinear elliptic partial differential equation with a linear combination of Dirac measures as a forcing term; the control variable corresponds to the amplitude of such singular sources. We analyze the existence of optimal solutions and derive first- and, necessary and sufficient, second-order optimality conditions. We develop
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Unconditionally stable small stencil enriched multiple point flux approximations of heterogeneous diffusion problems on general meshes IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-11-25 Julien Coatléven
We derive new multiple point flux approximations (MPFA) for the finite volume approximation of heterogeneous and anisotropic diffusion problems on general meshes, in dimensions 2 and 3. The resulting methods are unconditionally stable while preserving the small stencil typical of MPFA finite volumes. The key idea is to solve local variational problems with a well-designed stabilization term from which
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Adaptive VEM for variable data: convergence and optimality IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-11-16 L Beirão da Veiga, C Canuto, R H Nochetto, G Vacca, M Verani
We design an adaptive virtual element method (AVEM) of lowest order over triangular meshes with hanging nodes in 2d, which are treated as polygons. AVEM hinges on the stabilization-free a posteriori error estimators recently derived in Beirão da Veiga et al. (2023, Adaptive VEM: stabilization-free a posteriori error analysis and contraction property. SIAM J. Numer. Anal., 61, 457–494). The crucial
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Convergent finite element methods for the perfect conductivity problem with close-to-touching inclusions IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-11-14 Buyang Li, Haigang Li, Zongze Yang
In the perfect conductivity problem (i.e., the conductivity problem with perfectly conducting inclusions), the gradient of the electric field is often very large in a narrow region between two inclusions and blows up as the distance between the inclusions tends to zero. The rigorous error analysis for the computation of such perfect conductivity problems with close-to-touching inclusions of general
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A two-dimensional boundary value problem of elliptic type with nonlocal conjugation conditions IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-11-14 Zorica Milovanović Jeknić, Aleksandra Delić, Sandra Živanović
We consider an elliptic boundary value problem with nonlocal conjugation conditions. An a priori estimate for its weak solution in an appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate, compatible with the smoothness of the input data, is obtained.
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Numerical analysis of a hybridized discontinuous Galerkin method for the Cahn–Hilliard problem IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-11-11 Keegan L A Kirk, Beatrice Riviere, Rami Masri
The mixed form of the Cahn–Hilliard equations is discretized by the hybridized discontinuous Galerkin method. For any chemical energy density, existence and uniqueness of the numerical solution is obtained. The scheme is proved to be unconditionally stable. Convergence of the method is obtained by deriving a priori error estimates that are valid for the Ginzburg–Landau chemical energy density and for
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Two-scale methods for the normalized infinity Laplacian: rates of convergence IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-11-11 Wenbo Li, Abner J Salgado
We propose a monotone and consistent numerical scheme for the approximation of the Dirichlet problem for the normalized infinity Laplacian, which could be related to the family of the so-called two-scale methods. We show that this method is convergent and prove rates of convergence. These rates depend not only on the regularity of the solution, but also on whether or not the right-hand side vanishes
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Weak error analysis for strong approximation schemes of SDEs with super-linear coefficients IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-11-11 Xiaojie Wang, Yuying Zhao, Zhongqiang Zhang
We present an error analysis of weak convergence of one-step numerical schemes for stochastic differential equations (SDEs) with super-linearly growing coefficients. Following Milstein’s weak error analysis on the one-step approximation of SDEs, we prove a general result on weak convergence of the one-step discretization of the SDEs mentioned above. As applications, we show the weak convergence rates
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Uniform L∞-bounds for energy-conserving higher-order time integrators for the Gross–Pitaevskii equation with rotation IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-11-07 Christian Döding, Patrick Henning
In this paper, we consider an energy-conserving continuous Galerkin discretization of the Gross–Pitaevskii equation with a magnetic trapping potential and a stirring potential for angular momentum rotation. The discretization is based on finite elements in space and time and allows for arbitrary polynomial orders. It was first analyzed by O. Karakashian and C. Makridakis (SIAM J. Numer. Anal., 36(6)
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Convergent evolving finite element approximations of boundary evolution under shape gradient flow IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-10-30 Wei Gong, Buyang Li, Qiqi Rao
As a specific type of shape gradient descent algorithm, shape gradient flow is widely used for shape optimization problems constrained by partial differential equations. In this approach, the constraint partial differential equations could be solved by finite element methods on a domain with a solution-driven evolving boundary. Rigorous analysis for the stability and convergence of such finite element
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Homogeneous multigrid for HDG applied to the Stokes equation IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-10-24 Peipei Lu, Wei Wang, Guido Kanschat, Andreas Rupp
We propose a multigrid method to solve the linear system of equations arising from a hybrid discontinuous Galerkin (in particular, a single face hybridizable, a hybrid Raviart–Thomas, or a hybrid Brezzi–Douglas–Marini) discretization of a Stokes problem. Our analysis is centered around the augmented Lagrangian approach and we prove uniform convergence in this setting. Beyond this, we establish relations
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Higher-order adaptive methods for exit times of Itô diffusions IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-10-19 Håkon Hoel, Sankarasubramanian Ragunathan
We construct a higher-order adaptive method for strong approximations of exit times of Itô stochastic differential equations (SDEs). The method employs a strong Itô–Taylor scheme for simulating SDE paths, and adaptively decreases the step size in the numerical integration as the solution approaches the boundary of the domain. These techniques complement each other nicely: adaptive timestepping improves
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A local energy-based discontinuous Galerkin method for fourth-order semilinear wave equations IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-10-16 Lu Zhang
This paper proposes an energy-based discontinuous Galerkin scheme for fourth-order semilinear wave equations, which we rewrite as a system of second-order spatial derivatives. Compared to the local discontinuous Galerkin methods, the proposed scheme uses fewer auxiliary variables and is more computationally efficient. We prove several properties of the scheme. For example, we show that the scheme is
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Mixed Virtual Element approximation of linear acoustic wave equation IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-10-14 Franco Dassi, Alessio Fumagalli, Ilario Mazzieri, Giuseppe Vacca
We design a Mixed Virtual Element Method for the approximated solution to the first-order form of the acoustic wave equation. In the absence of external loads, the semi-discrete method exactly conserves the system energy. To integrate in time the semi-discrete problem we consider a classical $\theta $-method scheme. We carry out the stability and convergence analysis in the energy norm for the semi-discrete
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Convergence of Lagrange finite element methods for Maxwell eigenvalue problem in 3D IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-10-14 Daniele Boffi, Sining Gong, Johnny Guzmán, Michael Neilan
We prove convergence of the Maxwell eigenvalue problem using quadratic or higher Lagrange finite elements on Worsey–Farin splits in three dimensions. To do this, we construct two Fortin-like operators to prove uniform convergence of the corresponding source problem. We present numerical experiments to illustrate the theoretical results.
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A new step size selection strategy for the superiorization methodology using subgradient vectors and its application for solving convex constrained optimization problems IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-10-10 Mokhtar Abbasi, Mahdi Ahmadinia, Ali Ahmadinia
This paper presents a novel approach for solving convex constrained minimization problems by introducing a special subclass of quasi-nonexpansive operators and combining them with the superiorization methodology that utilizes subgradient vectors. Superiorization methodology tries to reduce a target function while seeking a feasible point for the given constraints. We begin by introducing a new class
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Milstein schemes and antithetic multilevel Monte Carlo sampling for delay McKean–Vlasov equations and interacting particle systems IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-09-06 Jianhai Bao, Christoph Reisinger, Panpan Ren, Wolfgang Stockinger
In this paper, we first derive Milstein schemes for an interacting particle system associated with point delay McKean–Vlasov stochastic differential equations, possibly with a drift term exhibiting super-linear growth in the state component. We prove strong convergence of order one and moment stability, making use of techniques from variational calculus on the space of probability measures with finite
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Discrete Gagliardo–Nirenberg inequality and application to the finite volume approximation of a convection–diffusion equation with a Joule effect term IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-08-30 Caterina Calgaro, Clément Cancès, Emmanuel Creusé
A discrete order-two Gagliardo–Nirenberg inequality is established for piecewise constant functions defined on a two-dimensional structured mesh composed of rectangular cells. As in the continuous framework, this discrete Gagliardo–Nirenberg inequality allows to control in particular the $L^4$ norm of the discrete gradient of the numerical solution by the $L^2$ norm of its discrete Hessian times its
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An arbitrary-order discrete rot-rot complex on polygonal meshes with application to a quad-rot problem IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-08-29 Daniele A Di Pietro
In this work, following the discrete de Rham approach, we develop a discrete counterpart of a two-dimensional de Rham complex with enhanced regularity. The proposed construction supports general polygonal meshes and arbitrary approximation orders. We establish exactness on a contractible domain for both the versions of the complex with and without boundary conditions and, for the former, prove a complete
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A nodally bound-preserving finite element method IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-08-26 Gabriel R Barrenechea, Emmanuil H Georgoulis, Tristan Pryer, Andreas Veeser
This work proposes a nonlinear finite element method whose nodal values preserve bounds known for the exact solution. The discrete problem involves a nonlinear projection operator mapping arbitrary nodal values into bound-preserving ones and seeks the numerical solution in the range of this projection. As the projection is not injective, a stabilisation based upon the complementary projection is added
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Numerical approximation of singular-degenerate parabolic stochastic partial differential equations IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-08-25 Ľubomír Baňas, Benjamin Gess, Christian Vieth
We study a general class of singular degenerate parabolic stochastic partial differential equations (SPDEs) that include, in particular, the stochastic porous medium equations and the stochastic fast diffusion equation. We propose a fully discrete numerical approximation of the considered SPDEs based on the very weak formulation. By exploiting the monotonicity properties of the proposed formulation
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High order approximations of the Cox–Ingersoll–Ross process semigroup using random grids IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-08-25 Aurélien Alfonsi, Edoardo Lombardo
We present new high order approximations schemes for the Cox–Ingersoll–Ross (CIR) process that are obtained by using a recent technique developed by Alfonsi and Bally (2021, A generic construction for high order approximation schemes of semigroups using random grids. Numer. Math., 148, 743–793) for the approximation of semigroups. The idea consists in using a suitable combination of discretization
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A second-order bulk–surface splitting for parabolic problems with dynamic boundary conditions IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-08-12 Robert Altmann, Christoph Zimmer
This paper introduces a novel approach for the construction of bulk–surface splitting schemes for semilinear parabolic partial differential equations with dynamic boundary conditions. The proposed construction is based on a reformulation of the system as a partial differential–algebraic equation and the inclusion of certain delay terms for the decoupling. To obtain a fully discrete scheme, the splitting
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Optimized Schwarz methods for the time-dependent Stokes–Darcy coupling IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-08-05 Marco Discacciati, Tommaso Vanzan
This paper derives optimized coefficients for optimized Schwarz iterations for the time-dependent Stokes–Darcy problem using an innovative strategy to solve a nonstandard min-max problem. The coefficients take into account both physical and discretization parameters that characterize the coupled problem, and they guarantee the robustness of the associated domain decomposition method. Numerical results
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On symmetric positive definite preconditioners for multiple saddle-point systems IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-08-05 John W Pearson, Andreas Potschka
We consider symmetric positive definite preconditioners for multiple saddle-point systems of block tridiagonal form, which can be applied within the Minres algorithm. We describe such a preconditioner for which the preconditioned matrix has only two distinct eigenvalues, $1$ and $-1$, when the preconditioner is applied exactly. We discuss the relative merits of such an approach compared to a more widely
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A posteriori error analysis of space-time discontinuous Galerkin methods for the ε-stochastic Allen–Cahn equation IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-08-03 Dimitra C Antonopoulou, Bernard Egwu, Yubin Yan
In this work, we apply an a posteriori error analysis for the space-time, discontinuous in time, Galerkin scheme, which has been proposed in Antonopoulou (2020, Space-time discontinuous Galerkin methods for the $\varepsilon $-dependent stochastic Allen–Cahn equation with mild noise. IMA J. Num. Analysis, 40, 2076–2105) for the $\varepsilon $-dependent stochastic Allen–Cahn equation with mild noise
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Optimal numerical integration and approximation of functions on ℝd equipped with Gaussian measure IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-08-03 Dinh Dũng, Van Kien Nguyen
We investigate the numerical approximation of integrals over $\mathbb{R}^{d}$ equipped with the standard Gaussian measure $\gamma $ for integrands belonging to the Gaussian-weighted Sobolev spaces $W^{\alpha }_{p}(\mathbb{R}^{d}, \gamma )$ of mixed smoothness $\alpha \in \mathbb{N}$ for $1 < p < \infty $. We prove the asymptotic order of the convergence of optimal quadratures based on $n$ integration
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The Gaussian wave packet transform via quadrature rules IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-07-29 Paul Bergold, Caroline Lasser
We analyse the Gaussian wave packet transform. Based on the Fourier inversion formula and a partition of unity, which is formed by a collection of Gaussian basis functions, a new representation of square-integrable functions is presented. Including a rigorous error analysis, the variants of the wave packet transform are then derived by a discretization of the Fourier integral via different quadrature
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On the reduction in accuracy of finite difference schemes on manifolds without boundary IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-07-20 Brittany Froese Hamfeldt, Axel G R Turnquist
We investigate error bounds for numerical solutions of divergence structure linear elliptic partial differential equations (PDEs) on compact manifolds without boundary. Our focus is on a class of monotone finite difference approximations, which provide a strong form of stability that guarantees the existence of a bounded solution. In many settings including the Dirichlet problem, it is easy to show
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Convergence guarantees for coefficient reconstruction in PDEs from boundary measurements by variational and Newton-type methods via range invariance IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-07-20 Barbara Kaltenbacher
A key observation underlying this paper is the fact that the range invariance condition for convergence of regularization methods for nonlinear ill-posed operator equations—such as coefficient identification in partial differential equations (PDEs) from boundary observations—can often be achieved by extending the searched for parameter in the sense of allowing it to depend on additional variables.
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Adapting the centred simplex gradient to compensate for misaligned sample points IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-07-19 Yiwen Chen, Warren Hare
The centred simplex gradient (CSG) is a popular gradient approximation technique in derivative-free optimization. Its computation requires a perfectly symmetric set of sample points and is known to provide an accuracy of $\mathcal {O}(\varDelta ^2)$, where $\varDelta $ is the radius of the sampling set. In this paper, we consider the situation where the set of sample points is not perfectly symmetric
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Adaptive interior penalty hybridized discontinuous Galerkin methods for Darcy flow in fractured porous media IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-07-19 Haitao Leng, Huangxin Chen
In this paper, we design and analyze an interior penalty hybridized discontinuous Galerkin (IP-HDG) method for the Darcy flow in the two- and three-dimensional fractured porous media. The discrete fracture model is used to model the fractures. The piecewise polynomials of degree $k$ are employed to approximate the pressure in the fractures and the pressure in the surrounding porous media. We prove
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A unified approach to maximum-norm a posteriori error estimation for second-order time discretizations of parabolic equations IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-07-05 Torsten Linβ, Martin Ossadnik, Goran Radojev
A class of linear parabolic equations is considered. We derive a common framework for the a posteriori error analysis of certain second-order time discretizations combined with finite element discretizations in space. In particular, we study the Crank–Nicolson method, the extrapolated Euler method, the backward differentiation formula of order 2, the Lobatto IIIC method and a two-stage SDIRK method
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Error analysis of time-discrete random batch method for interacting particle systems and associated mean-field limits IMA J. Numer. Anal. (IF 2.1) Pub Date : 2023-06-24 Xuda Ye, Zhennan Zhou
The random batch method provides an efficient algorithm for computing statistical properties of a canonical ensemble of interacting particles. In this work, we study the error estimates of the fully discrete random batch method, especially in terms of approximating the invariant distribution. The triangle inequality framework employed in this paper is a convenient approach to estimate the long-time