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Efficient low rank matrix recovery with flexible group sparse regularization IMA J. Numer. Anal. (IF 2.3) Pub Date : 2025-01-30 Quan Yu, Minru Bai, Xinzhen Zhang
In this paper, we present a novel approach to the low rank matrix recovery (LRMR) problem by casting it as a group sparsity problem. Specifically, we propose a flexible group sparse regularizer (FLGSR) that can group any number of matrix columns as a unit, whereas existing methods group each column as a unit. We prove the equivalence between the matrix rank and the FLGSR under some mild conditions
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Gauss quadrature rules for integrals involving weight functions with variable exponents and an application to weakly singular Volterra integral equations IMA J. Numer. Anal. (IF 2.3) Pub Date : 2025-01-23 Chafik Allouch, Gradimir V Milovanović
This paper presents a numerical integration approach that can be used to approximate on a finite interval, the integrals of functions that contain Jacobi weights with variable exponents. A modification of the integrand close to the singularities is needed, and a new modification is proposed. An application of such a rule to the numerical solution of variable-exponent weakly singular Volterra integral
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Compound Poisson particle approximation for McKean–Vlasov SDEs IMA J. Numer. Anal. (IF 2.3) Pub Date : 2025-01-23 Xicheng Zhang
We present a comprehensive discretization scheme for linear and nonlinear stochastic differential equations (SDEs) driven by either Brownian motions or $\alpha $-stable processes. Our approach utilizes compound Poisson particle approximations, allowing for simultaneous discretization of both the time and space variables in McKean–Vlasov SDEs. Notably, the approximation processes can be represented
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Numerical analysis of a spherical harmonic discontinuous Galerkin method for scaled radiative transfer equations with isotropic scattering IMA J. Numer. Anal. (IF 2.3) Pub Date : 2025-01-23 Qiwei Sheng, Cory D Hauck, Yulong Xing
In highly diffusion regimes when the mean free path $\varepsilon $ tends to zero, the radiative transfer equation has an asymptotic behavior which is governed by a diffusion equation and the corresponding boundary condition. Generally, a numerical scheme for solving this problem has the truncation error containing an $\varepsilon ^{-1}$ contribution that leads to a nonuniform convergence for small
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A fast algorithm for smooth convex minimization problems and its application to inverse source problems IMA J. Numer. Anal. (IF 2.3) Pub Date : 2025-01-21 Pham Quy Muoi, Vo Quang Duy, Chau Vinh Khanh, Nguyen Trung Thành
In this paper, we propose a fast algorithm for smooth convex minimization problems in a real Hilbert space whose objective functionals have Lipschitz continuous Fréchet derivatives. The main advantage of the proposed algorithm is that it has the optimal-order convergence rate and faster than Nesterov’s algorithm with the best setting. To demonstrate the efficiency of the proposed algorithm, we compare
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Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems IMA J. Numer. Anal. (IF 2.3) Pub Date : 2025-01-19 Lise-Marie Imbert-Gérard, Andrea Moiola, Chiara Perinati, Paul Stocker
Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying partial differential equation (PDE). Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous and have piecewise-constant coefficients. However, if the equation has variable coefficients, exact solutions are generally unavailable. Quasi-Trefftz
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Discontinuous Galerkin discretization of coupled poroelasticity–elasticity problems IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-12-28 Paola F Antonietti, Michele Botti, Ilario Mazzieri
This work is concerned with the analysis of a space–time finite element discontinuous Galerkin method on polytopal meshes (XT-PolydG) for the numerical discretization of wave propagation in coupled poroelastic–elastic media. The mathematical model consists of the low-frequency Biot’s equations in the poroelastic medium and the elastodynamics equation for the elastic one. To realize the coupling suitable
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Optimal error analysis of the normalized tangent plane FEM for Landau–Lifshitz–Gilbert equation IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-12-28 Rong An, Yonglin Li, Weiwei Sun
The dynamics of the magnetization in ferromagnetic materials is governed by the Landau–Lifshitz–Gilbert equation, which is highly nonlinear with the nonconvex sphere constraint $|{\textbf{m}}|=1$. A crucial issue in designing numerical schemes is to preserve this sphere constraint in the discrete level. A popular numerical method is the normalized tangent plane finite element method (NTP-FEM), which
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Parametric finite-element discretization of the surface Stokes equations: inf-sup stability and discretization error analysis IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-12-26 Hanne Hardering, Simon Praetorius
We study a higher-order surface finite-element penalty-based discretization of the tangential surface Stokes problem. Several discrete formulations are investigated, which are equivalent in the continuous setting. The impact of the choice of discretization of the diffusion term and of the divergence term on numerical accuracy and convergence, as well as on implementation advantages, is discussed. We
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Convergence and quasi-optimality of an AFEM for the Dirichlet boundary control problem IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-12-26 Arnab Pal, Thirupathi Gudi
In this article, convergence and quasi-optimal rate of convergence of an Adaptive Finite Element Method is shown for the Dirichlet boundary control problem that was proposed by Chowdhury et al. (2017, Error bounds for a Dirichlet boundary control problem based on energy spaces, Math. Comp., 86, 1103–1126). The theoretical results are illustrated by numerical experiments.
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A conforming multi-domain Legendre spectral method for solving diffusive-viscous wave equations in the exterior domain with separated star-shaped obstacles IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-12-14 Guoqing Yao, Zicheng Wang, Zhongqing Wang
In this paper, we propose a conforming multi-domain spectral method that combines mapping techniques to solve the diffusive-viscous wave equation in the exterior domain of two complex obstacles. First, we confine the exterior domain within a relatively large rectangular computational domain. Then, we decompose the rectangular domain into two sub-domains, each containing one obstacle. By applying coordinate
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The Milstein scheme for singular SDEs with Hölder continuous drift IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-12-14 Máté Gerencsér, Gerald Lampl, Chengcheng Ling
We study the $L^{p}$ rate of convergence of the Milstein scheme for stochastic differential equations when the drift coefficients possess only Hölder regularity. If the diffusion is elliptic and sufficiently regular, we obtain rates consistent with the additive case. The proof relies on regularization by noise techniques, particularly stochastic sewing, which in turn requires (at least asymptotically)
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Asymptotic consistency of the WSINDy algorithm in the limit of continuum data IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-12-13 Daniel A Messenger, David M Bortz
In this work we study the asymptotic consistency of the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) in the identification of differential equations from noisy samples of solutions. We prove that the WSINDy estimator is unconditionally asymptotically consistent for a wide class of models that includes the Navier–Stokes, Kuramoto–Sivashinsky and Sine–Gordon equations. We
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A spectral collocation method for functional and delay differential equations IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-11-29 Nicholas Hale
A framework for Chebyshev spectral collocation methods for the numerical solution of functional and delay differential equations (FDEs and DDEs) is described. The framework combines interpolation via the barycentric resampling matrix with a multidomain approach used to resolve isolated discontinuities propagated by nonsmooth initial data. Geometric convergence in the number of degrees of freedom is
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Error analysis for a finite element approximation of the steady p·-Navier–Stokes equations IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-11-25 Luigi C Berselli, Alex Kaltenbach
In this paper, we examine a finite element approximation of the steady $p(\cdot )$-Navier–Stokes equations ($p(\cdot )$ is variable dependent) and prove orders of convergence by assuming natural fractional regularity assumptions on the velocity vector field and the kinematic pressure. Compared to previous results, we treat the convective term and employ a more practicable discretization of the power-law
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A unified framework for the error analysis of physics-informed neural networks IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-11-20 Marius Zeinhofer, Rami Masri, Kent–André Mardal
We prove a priori and a posteriori error estimates for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations, and a PDE constrained optimization problem. For the analysis, we propose an abstract framework in the common language of bilinear forms, and we show that coercivity and continuity
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On the maximum bound principle and energy dissipation of exponential time differencing methods for the matrix-valued Allen–Cahn equation IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-11-19 Yaru Liu, Chaoyu Quan, Dong Wang
This work delves into the exponential time differencing (ETD) schemes for the matrix-valued Allen–Cahn equation. In fact, the maximum bound principle (MBP) for the first- and second-order ETD schemes is presented in a prior publication [SIAM Review, 63(2), 2021], assuming a symmetric initial matrix field. Noteworthy is our novel contribution, demonstrating that the first- and second-order ETD schemes
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The weighted and shifted seven-step BDF method for parabolic equations IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-11-18 Georgios Akrivis, Minghua Chen, Fan Yu
Stability of the BDF methods of order up to 5 for parabolic equations can be established by the energy technique via Nevanlinna–Odeh multipliers. The nonexistence of Nevanlinna–Odeh multipliers makes the six-step BDF method special; however, the energy technique was recently extended by the authors in Akrivis et al. (2021, SIAM J. Numer. Anal., 59, 2449–2472) and covers all six stable BDF methods.
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Stability estimates of Nyström discretizations of Helmholtz decomposition boundary integral equation formulations for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-11-10 Víctor Domínguez, Catalin Turc
Helmholtz decompositions of elastic fields is a common approach for the solution of Navier scattering problems. Used in the context of boundary integral equations (BIE), this approach affords solutions of Navier problems via the simpler Helmholtz boundary integral operators (BIOs). Approximations of Helmholtz Dirichlet-to-Neumann (DtN) can be employed within a regularizing combined field strategy to
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Positive definite functions on a regular domain IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-11-06 Martin Buhmann, Yuan Xu
We define positive and strictly positive definite functions on a domain and study these functions on a list of regular domains. The list includes the unit ball, conic surface, hyperbolic surface, solid hyperboloid and the simplex. Each of these domains is embedded in a quadrant or a union of quadrants of the unit sphere by a distance-preserving map, from which characterizations of positive definite
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Fast time-stepping discontinuous Galerkin method for the subdiffusion equation IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-10-31 Hui Zhang, Fanhai Zeng, Xiaoyun Jiang, Zhimin Zhang
The nonlocality of the fractional operator causes numerical difficulties for long time computation for time-fractional evolution equations. This paper develops a high-order fast time-stepping discontinuous Galerkin (DG) finite element method for a time-fractional diffusion equation, which saves storage and computational time. An optimal error estimate of the form $O(N^{-p-1} + h^{m+1} + \varepsilon
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An extension of the approximate component mode synthesis method to the heterogeneous Helmholtz equation IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-10-26 Elena Giammatteo, Alexander Heinlein, Matthias Schlottbom
In this work, we propose and analyze an extension of the approximate component mode synthesis (ACMS) method to the two-dimensional heterogeneous Helmholtz equation. The ACMS method has originally been introduced by Hetmaniuk and Lehoucq as a multiscale method to solve elliptic partial differential equations. The ACMS method uses a domain decomposition to separate the numerical approximation by splitting
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Time-dependent electromagnetic scattering from dispersive materials IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-10-25 Jörg Nick, Selina Burkhard, Christian Lubich
This paper studies time-dependent electromagnetic scattering from obstacles that are described by dispersive material laws. We consider the numerical treatment of a scattering problem in which a dispersive material law, for a causal and passive homogeneous material, determines the wave–material interaction in the scatterer. The resulting problem is nonlocal in time inside the scatterer and is posed
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An exponential stochastic Runge–Kutta type method of order up to 1.5 for SPDEs of Nemytskii-type IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-10-17 Claudine von Hallern, Ricarda Missfeldt, Andreas Rössler
For the approximation of solutions for stochastic partial differential equations, numerical methods that obtain a high order of convergence and at the same time involve reasonable computational cost are of particular interest. We therefore propose a new numerical method of exponential stochastic Runge–Kutta type that allows for convergence with a temporal order of up to $\frac{3}/{2}$ and that can
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Three types of quasi-Trefftz functions for the 3D convected Helmholtz equation: construction and approximation properties IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-10-10 Lise-Marie Imbert-Gérard, Guillaume Sylvand
Trefftz methods are numerical methods for the approximation of solutions to boundary and/or initial value problems. They are Galerkin methods with particular test and trial functions, which solve locally the governing partial differential equation (PDE). This property is called the Trefftz property. Quasi-Trefftz methods were introduced to leverage the advantages of Trefftz methods for problems governed
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A convergent stochastic scalar auxiliary variable method IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-10-10 Stefan Metzger
We discuss an extension of the scalar auxiliary variable approach, which was originally introduced by Shen et al. (2018, The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys., 353, 407–416) for the discretization of deterministic gradient flows. By introducing an additional scalar auxiliary variable this approach allows to derive a linear scheme while still maintaining unconditional
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A priori and a posteriori error analysis of a mixed DG method for the three-field quasi-Newtonian Stokes flow IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-10-03 Lina Zhao
In this paper we propose and analyse a new mixed-type DG method for the three-field quasi-Newtonian Stokes flow. The scheme is based on the introduction of the stress and strain tensor as further unknowns as well as the elimination of the pressure variable by means of the incompressibility constraint. As such, the resulting system involves three unknowns: the stress, the strain tensor and the velocity
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Discrete maximum-minimum principle for a linearly implicit scheme for nonlinear parabolic FEM problems under weakened time restrictions IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-09-28 István Faragó, Róbert Horváth, János Karátson
In this paper, we extend our earlier results in Faragó, I., Karátson, J. and Korotov, S. (2012, Discrete maximum principles for nonlinear parabolic PDE systems. IMA J. Numer. Anal., 32, 1541–1573) on the discrete maximum-minimum principle (DMP) for nonlinear parabolic systems of PDEs. We propose a linearly implicit scheme, where only linear problems have to be solved on the time layers. We obtain a
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An unconditionally energy dissipative, adaptive IMEX BDF2 scheme and its error estimates for Cahn–Hilliard equation on generalized SAV approach IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-09-25 Yifan Wei, Jiwei Zhang, Chengchao Zhao, Yanmin Zhao
An adaptive implicit-explicit (IMEX) BDF2 scheme is investigated on generalized SAV approach for the Cahn–Hilliard equation by combining with Fourier spectral method in space. It is proved that the modified energy dissipation law is unconditionally preserved at discrete levels. Under a mild ratio restriction, i.e., A1: $0
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A higher order multiscale method for the wave equation IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-09-19 Felix Krumbiegel, Roland Maier
In this paper we propose a multiscale method for the acoustic wave equation in highly oscillatory media. We use a higher order extension of the localized orthogonal decomposition method combined with a higher order time stepping scheme and present rigorous a priori error estimates in the energy-induced norm. We find that in the very general setting without additional assumptions on the coefficient
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High-order energy stable discrete variational derivative schemes for gradient flows IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-09-19 Jizu Huang
The existing discrete variational derivative method is fully implicit and only second-order accurate for gradient flow. In this paper, we propose a framework to construct high-order implicit (original) energy stable schemes and second-order semi-implicit (modified) energy stable schemes. Combined with the Runge–Kutta process, we can build high-order and unconditionally (original) energy stable schemes
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A mesh-independent method for second-order potential mean field games IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-09-18 Kang Liu, Laurent Pfeiffer
This article investigates the convergence of the Generalized Frank–Wolfe (GFW) algorithm for the resolution of potential and convex second-order mean field games. More specifically, the impact of the discretization of the mean-field-game system on the effectiveness of the GFW algorithm is analyzed. The article focuses on the theta-scheme introduced by the authors in a previous study. A sublinear and
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A second-order correction method for loosely coupled discretizations applied to parabolic–parabolic interface problems IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-09-14 Erik Burman, Rebecca Durst, Miguel A Fernández, Johnny Guzmán, Sijing Liu
We consider a parabolic–parabolic interface problem and construct a loosely coupled prediction-correction scheme based on the Robin–Robin splitting method analyzed in [J. Numer. Math., 31(1):59–77, 2023]. We show that the errors of the correction step converge at $\mathcal O((\varDelta t)^{2})$, under suitable convergence rate assumptions on the discrete time derivative of the prediction step, where
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Numerical analysis of an evolving bulk–surface model of tumour growth IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-09-13 Dominik Edelmann, Balázs Kovács, Christian Lubich
This paper studies an evolving bulk–surface finite element method for a model of tissue growth, which is a modification of the model of Eyles, King and Styles (2019, A tractable mathematical model for tissue growth. Interfaces Free Bound, 21, 463–493). The model couples a Poisson equation on the domain with a forced mean curvature flow of the free boundary, with nontrivial bulk–surface coupling in
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Low regularity error estimates for the time integration of 2D NLS IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-09-13 Lun Ji, Alexander Ostermann, Frédéric Rousset, Katharina Schratz
A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the two-dimensional torus $\mathbb{T}^{2}$. The scheme is analysed in a framework of discrete Bourgain spaces, which allows us to consider initial data with low regularity; more precisely initial data in $H^{s}(\mathbb{T}^{2})$ with $s>0$. In this way, the usual stability restriction
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A mini immersed finite element method for two-phase Stokes problems on Cartesian meshes IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-09-09 Haifeng Ji, Dong Liang, Qian Zhang
This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element, with shape functions modified on interface elements according to interface jump conditions while keeping the degrees of freedom unchanged. Both discontinuous viscosity coefficients and surface
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New Banach spaces-based mixed finite element methods for the coupled poroelasticity and heat equations IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-09-05 Julio Careaga, Gabriel N Gatica, Cristian Inzunza, Ricardo Ruiz-Baier
In this paper, we introduce and analyze a Banach spaces-based approach yielding a fully-mixed finite element method for numerically solving the coupled poroelasticity and heat equations, which describe the interaction between the fields of deformation and temperature. A nonsymmetric pseudostress tensor is utilized to redefine the constitutive equation for the total stress, which is an extension of
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A posteriori error analysis of a positivity preserving scheme for the power-law diffusion Keller–Segel model IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-08-28 Jan Giesselmann, Niklas Kolbe
We study a finite volume scheme approximating a parabolic-elliptic Keller–Segel system with power-law diffusion with exponent $\gamma \in [1,3]$ and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the $L^{\infty }(0,T;H^{1}(\varOmega ))$ norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability
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Necessary and sufficient conditions for avoiding Babuška’s paradox on simplicial meshes IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-08-27 Sören Bartels, Philipp Tscherner
It is shown that discretizations based on variational or weak formulations of the plate bending problem with simple support boundary conditions do not lead to the failure of convergence when polygonal domain approximations are used and the imposed boundary conditions are compatible with the nodal interpolation of the restriction of certain regular functions to approximating domains. It is further shown
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Computing Klein-Gordon Spectra IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-08-26 Frank Rösler, Christiane Tretter
We study the computational complexity of the eigenvalue problem for the Klein–Gordon equation in the framework of the Solvability Complexity Index Hierarchy. We prove that the eigenvalue of the Klein–Gordon equation with linearly decaying potential can be computed in a single limit with guaranteed error bounds from above. The proof is constructive, i.e. we obtain a numerical algorithm that can be implemented
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Finite element analysis of a generalized Robin boundary value problem in curved domains based on the extension approach IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-08-23 Takahito Kashiwabara
A theoretical analysis of the finite element method for a generalized Robin boundary value problem, which involves a second-order differential operator on the boundary, is presented. If $\varOmega $ is a general smooth domain with a curved boundary, we need to introduce an approximate domain $\varOmega _{h}$ and to address issues owing to the domain perturbation $\varOmega \neq \varOmega _{h}$. In
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Mixed finite elements for the Gross–Pitaevskii eigenvalue problem: a priori error analysis and guaranteed lower energy bound IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-08-22 Dietmar Gallistl, Moritz Hauck, Yizhou Liang, Daniel Peterseim
We establish an a priori error analysis for the lowest-order Raviart–Thomas finite element discretization of the nonlinear Gross-Pitaevskii eigenvalue problem. Optimal convergence rates are obtained for the primal and dual variables as well as for the eigenvalue and energy approximations. In contrast to conforming approaches, which naturally imply upper energy bounds, the proposed mixed discretization
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Discrete hypocoercivity for a nonlinear kinetic reaction model IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-08-19 Marianne Bessemoulin-Chatard, Tino Laidin, Thomas Rey
In this article we propose a finite-volume discretization of a one-dimensional nonlinear reaction kinetic model proposed in Neumann & Schmeiser (2016), which describes a two-species recombination-generation process. Specifically, we establish the long-time convergence of approximate solutions towards equilibrium, at exponential rate. The study is based on an adaptation for a discretization of the linearized
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Convergence of an adaptive C0-interior penalty Galerkin method for the biharmonic problem IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-08-19 Alexander Dominicus, Fernando D Gaspoz, Christian Kreuzer
We develop a basic convergence analysis for an adaptive ${C}^{0}\mathsf{IPG}$ method for the biharmonic problem that provides convergence without rates for all practically relevant marking strategies and all penalty parameters assuring coercivity of the method. The analysis hinges on embedding properties of (broken) Sobolev and BV spaces, and the construction of a suitable limit space. In contrast
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On the linear convergence of additive Schwarz methods for the p-Laplacian IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-08-15 Young-Ju Lee, Jongho Park
We consider additive Schwarz methods for boundary value problems involving the $p$-Laplacian. While existing theoretical estimates suggest a sublinear convergence rate for these methods, empirical evidence from numerical experiments demonstrates a linear convergence rate. In this paper we narrow the gap between these theoretical and empirical results by presenting a novel convergence analysis. First
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Mass, momentum and energy preserving FEEC and broken-FEEC schemes for the incompressible Navier–Stokes equations IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-08-15 Valentin Carlier, Martin Campos Pinto, Francesco Fambri
In this article we propose two finite-element schemes for the Navier–Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the point-wise divergence
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MultiShape: a spectral element method, with applications to Dynamic Density Functional Theory and PDE-constrained optimization IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-08-14 Jonna C Roden, Rory D Mills-Williams, John W Pearson, Benjamin D Goddard
A new numerical framework is developed to solve general nonlinear and nonlocal PDEs on complicated two-dimensional domains. This enables the solution of a wide range of both steady and time-dependent problems on nonstandard geometries, as well as providing the ability to impose nonlinear and nonlocal boundary conditions (typical of those arising in the modelling of physical phenomena) in a flexible
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Numerical method and error estimate for stochastic Landau–Lifshitz–Bloch equation IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-08-10 Beniamin Goldys, Chunxi Jiao, Kim-Ngan Le
In this paper we study numerical methods for solving a system of quasilinear stochastic partial differential equations known as the stochastic Landau–Lifshitz–Bloch (LLB) equation on a bounded domain in ${\mathbb{R}}^{d}$ for $d=1,2$. Our main results are estimates of the rate of convergence of the Finite Element Method to the solutions of stochastic LLB. To overcome the lack of regularity of the solution
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On the fast convergence of minibatch heavy ball momentum IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-08-09 Raghu Bollapragada, Tyler Chen, Rachel Ward
Simple stochastic momentum methods are widely used in machine learning optimization, but their good practical performance is at odds with an absence of theoretical guarantees of acceleration in the literature. In this work, we aim to close the gap between theory and practice by showing that stochastic heavy ball momentum retains the fast linear rate of (deterministic) heavy ball momentum on quadratic
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Convergence rates under a range invariance condition with application to electrical impedance tomography IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-08-07 Barbara Kaltenbacher
This paper is devoted to proving convergence rates of variational and iterative regularization methods under variational source conditions variational source conditions (VSCs) for inverse problems whose linearization satisfies a range invariance condition. In order to achieve this, often an appropriate relaxation of the problem needs to be found that is usually based on an augmentation of the set of
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A coupled steady thermo-electromagnetic problem in axisymmetric geometries. Mathematical and numerical analysis IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-07-25 Dolores Gómez, Bibiana López-Rodríguez, Pilar Salgado, Pablo Venegas
This paper focuses on the analysis of a steady thermo-electromagnetic problem related to the modeling of induction heating processes. Taking advantage of the cylindrical symmetry, the original three-dimensional problem can be reduced to a two-dimensional one on a meridional section, provided that the current density has only the azimuthal component. A variational formulation is presented in appropriately
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Long-term accuracy of numerical approximations of SPDEs with the stochastic Navier–Stokes equations as a paradigm IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-07-16 Nathan E Glatt-Holtz, Cecilia F Mondaini
This work introduces a general framework for establishing the long time accuracy for approximations of Markovian dynamical systems on separable Banach spaces. Our results illuminate the role that a certain uniformity in Wasserstein contraction rates for the approximating dynamics bears on long time accuracy estimates. In particular, our approach yields weak consistency bounds on ${\mathbb{R}}^{+}$
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hp-optimal convergence of the original DG method for linear hyperbolic problems on special simplicial meshes IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-07-09 Z Dong, L Mascotto
We prove $hp$-optimal error estimates for the original discontinuous Galerkin (DG) method when approximating solutions to first-order hyperbolic problems with constant convection fields in the $L^{2}$ and DG norms. The main theoretical tools used in the analysis are novel $hp$-optimal approximation properties of the special projector introduced in Cockburn et al. (2008, Optimal convergence of the original
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The error bounds of Gaussian quadratures for one rational modification of Chebyshev measures IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-07-09 Rada M Mutavdžić Djukić
For an analytic integrand, the error term in the Gaussian quadrature can be represented as a contour integral, where the contour is commonly taken to be an ellipse. Thus, finding its upper bound can be reduced to finding the maximum of the modulus of the kernel on the ellipse. The location of this maximum was investigated in many special cases, particularly, for the Gaussian quadrature with respect
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Barycentric rational interpolation of exponentially clustered poles IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-07-06 Kelong Zhao, Shuhuang Xiang
We have developed a rational interpolation method for analytic functions with branch point singularities, which utilizes several exponentially clustered poles proposed by Trefethen and his collaborators (2021, Exponential node clustering at singularities for rational approximation, quadrature, and PDEs. Numer. Math., 147, 227–254). The key to the feasibility of this interpolation method is that the
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First-Order Perturbation Theory of Trust-Region Subproblem IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-07-06 Bo Feng, Gang Wu
Trust-region subproblem (TRS) is an important problem arising in many applications such as numerical optimization, Tikhonov regularization of ill-posed problems and constrained eigenvalue problems. In recent decades, extensive works focus on how to solve the trust-region subproblem efficiently. To the best of our knowledge, there are few results on perturbation analysis of the trust-region subproblem
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Continuous finite elements satisfying a strong discrete Miranda–Talenti identity IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-07-01 Dietmar Gallistl, Shudan Tian
This article introduces continuous $H^{2}$-nonconforming finite elements in two and three space dimensions that satisfy a strong discrete Miranda–Talenti inequality in the sense that the global $L^{2}$ norm of the piecewise Hessian is bounded by the $L^{2}$ norm of the piecewise Laplacian. The construction is based on globally continuous finite element functions with $C^{1}$ continuity on the vertices
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A geometric integration approach to smooth optimization: foundations of the discrete gradient method IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-07-01 Matthias J Ehrhardt, Erlend S Riis, Torbjørn Ringholm, Carola-Bibiane Schönlieb
Discrete gradient methods are geometric integration techniques that can preserve the dissipative structure of gradient flows. Due to the monotonic decay of the function values, they are well suited for general convex and nonconvex optimization problems. Both zero- and first-order algorithms can be derived from the discrete gradient method by selecting different discrete gradients. In this paper, we
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Interpolation of set-valued functions IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-06-25 Nira Dyn, David Levin, Qusay Muzaffar
Given a finite number of samples of a continuous set-valued function F, mapping an interval to compact subsets of the real line, we develop good approximations of F, which can be computed efficiently. In the first stage, we develop an efficient algorithm for computing an interpolant to $F$, inspired by the ‘metric polynomial interpolation’, which is based on the theory in Dyn et al. (2014, Approximation
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Asymptotically compatible energy of variable-step fractional BDF2 scheme for the time-fractional Cahn–Hilliard model IMA J. Numer. Anal. (IF 2.3) Pub Date : 2024-06-25 Hong-lin Liao, Nan Liu, Xuan Zhao
A novel discrete gradient structure of the variable-step fractional BDF2 formula approximating the Caputo fractional derivative of order $\alpha \in (0,1)$ is constructed by a local-nonlocal splitting technique, that is, the fractional BDF2 formula is split into a local part analogue to the two-step backward differentiation formula (BDF2) of the first derivative and a nonlocal part analogue to the