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Low regularity error estimates for the time integration of 2D NLS IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240913
Lun Ji, Alexander Ostermann, Frédéric Rousset, Katharina SchratzA filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the twodimensional 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

A mini immersed finite element method for twophase Stokes problems on Cartesian meshes IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240909
Haifeng Ji, Dong Liang, Qian ZhangThis paper presents a mini immersed finite element (IFE) method for solving two and threedimensional twophase 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

New Banach spacesbased mixed finite element methods for the coupled poroelasticity and heat equations IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240905
Julio Careaga, Gabriel N Gatica, Cristian Inzunza, Ricardo RuizBaierIn this paper, we introduce and analyze a Banach spacesbased approach yielding a fullymixed 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

Necessary and sufficient conditions for avoiding Babuška’s paradox on simplicial meshes IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240827
Sören Bartels, Philipp TschernerIt 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

Computing KleinGordon Spectra IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240826
Frank Rösler, Christiane TretterWe 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

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 : 20240822
Dietmar Gallistl, Moritz Hauck, Yizhou Liang, Daniel PeterseimWe establish an a priori error analysis for the lowestorder Raviart–Thomas finite element discretization of the nonlinear GrossPitaevskii 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

Mass, momentum and energy preserving FEEC and brokenFEEC schemes for the incompressible Navier–Stokes equations IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240815
Valentin Carlier, Martin Campos Pinto, Francesco FambriIn this article we propose two finiteelement 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 skewsymmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the pointwise divergence

Numerical method and error estimate for stochastic Landau–Lifshitz–Bloch equation IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240810
Beniamin Goldys, Chunxi Jiao, KimNgan LeIn 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

On the fast convergence of minibatch heavy ball momentum IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240809
Raghu Bollapragada, Tyler Chen, Rachel WardSimple 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

Longterm accuracy of numerical approximations of SPDEs with the stochastic Navier–Stokes equations as a paradigm IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240716
Nathan E GlattHoltz, Cecilia F MondainiThis 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 ${\mathbb{R}}^{+}$

The error bounds of Gaussian quadratures for one rational modification of Chebyshev measures IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240709
Rada M Mutavdžić DjukićFor 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

Barycentric rational interpolation of exponentially clustered poles IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240706
Kelong Zhao, Shuhuang XiangWe 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

FirstOrder Perturbation Theory of TrustRegion Subproblem IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240706
Bo Feng, Gang WuTrustregion subproblem (TRS) is an important problem arising in many applications such as numerical optimization, Tikhonov regularization of illposed problems and constrained eigenvalue problems. In recent decades, extensive works focus on how to solve the trustregion subproblem efficiently. To the best of our knowledge, there are few results on perturbation analysis of the trustregion subproblem

A geometric integration approach to smooth optimization: foundations of the discrete gradient method IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240701
Matthias J Ehrhardt, Erlend S Riis, Torbjørn Ringholm, CarolaBibiane SchönliebDiscrete 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 firstorder algorithms can be derived from the discrete gradient method by selecting different discrete gradients. In this paper, we

Interpolation of setvalued functions IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240625
Nira Dyn, David Levin, Qusay MuzaffarGiven a finite number of samples of a continuous setvalued 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

Asymptotically compatible energy of variablestep fractional BDF2 scheme for the timefractional Cahn–Hilliard model IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240625
Honglin Liao, Nan Liu, Xuan ZhaoA novel discrete gradient structure of the variablestep fractional BDF2 formula approximating the Caputo fractional derivative of order $\alpha \in (0,1)$ is constructed by a localnonlocal splitting technique, that is, the fractional BDF2 formula is split into a local part analogue to the twostep backward differentiation formula (BDF2) of the first derivative and a nonlocal part analogue to the

Convergence analysis for minimum action methods coupled with a finite difference method IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240621
Jialin Hong, Diancong Jin, Derui ShengThe minimum action method (MAM) is an effective approach to numerically solving minima and minimizers of Freidlin–Wentzell (FW) action functionals, which is used to study the most probable transition path and probability of the occurrence of transitions for stochastic differential equations (SDEs) with small noise. In this paper, we focus on MAMs based on a finite difference method with nonuniform

hpversion C1continuous Petrov–Galerkin method for nonlinear secondorder initial value problems with application to wave equations IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240621
Lina Wang, Mingzhu Zhang, Hongjiong Tian, Lijun YiWe introduce and analyze an $hp$version $C^{1}$continuous Petrov–Galerkin (CPG) method for nonlinear initial value problems of secondorder ordinary differential equations. We derive apriori error estimates in the $L^{2}$, $L^{\infty }$, $H^{1}$ and $H^{2}$norms that are completely explicit in the local time steps and local approximation degrees. Moreover, we show that the $hp$version $C^{1}$CPG

Precise error bounds for numerical approximations of fractional HJB equations IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240612
Indranil Chowdhury, Espen R JakobsenWe prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton–Jacobi–Bellman equations. We consider diffusioncorrected differencequadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations that are (formally) fractional order and secondorder methods. It is well known in numerical analysis that

CIPstabilized virtual elements for diffusionconvectionreaction problems IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240531
L Beirão da Veiga, C Lovadina, M TrezziThe Virtual Element Method (VEM) for diffusionconvectionreaction problems is considered. In order to design a quasirobust scheme also in the convectiondominated regime, a Continuous Interior Penalty approach is employed. Due to the presence of polynomial projection operators, typical of the VEM, the stability and the error analysis requires particular care—especially in treating the advective term

Stochastic modified equations for symplectic methods applied to rough Hamiltonian systems IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240524
Chuchu Chen, Jialin Hong, Chuying HuangWe investigate stochastic modified equations to explain the mathematical mechanism of symplectic methods applied to rough Hamiltonian systems. The contribution of this paper is threefold. First, we construct a new type of stochastic modified equation. For symplectic methods applied to rough Hamiltonian systems, the associated stochastic modified equations are proved to have Hamiltonian formulations

Discrete anisotropic curve shortening flow in higher codimension IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240524
Klaus Deckelnick, Robert NürnbergWe introduce a novel formulation for the evolution of parametric curves by anisotropic curve shortening flow in ${{\mathbb{R}}}^{d}$, $d\geq 2$. The reformulation hinges on a suitable manipulation of the parameterization’s tangential velocity, leading to a strictly parabolic differential equation. Moreover, the derived equation is in divergence form, giving rise to a natural variational numerical method

Highorder LagrangeGalerkin methods for the conservative formulation of the advectiondiffusion equation IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240518
Rodolfo Bermejo, Manuel ColeraWe introduce in this paper the numerical analysis of high order both in time and space LagrangeGalerkin methods for the conservative formulation of the advectiondiffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order $q=5$. The development and analysis of the methods are performed in the framework of time evolving finite elements presented in

An equilibrated estimator for mixed finite element discretizations of the curlcurl problem IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240517
T ChaumontFreletWe propose a new a posteriori error estimator for mixed finite element discretizations of the curlcurl problem. This estimator relies on a Prager–Synge inequality, and therefore leads to fully guaranteed constantfree upper bounds on the error. The estimator is also locally efficient and polynomialdegreerobust. The construction is based on patchwise divergenceconstrained minimization problems

Optimal convergence analysis of two RPCSAV schemes for the unsteady incompressible magnetohydrodynamics equations IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240515
Xiaojing Dong, Huayi Huang, Yunqing Huang, Xiaojuan Shen, Qili TangIn this paper, we present and analyze two linear and fully decoupled schemes for solving the unsteady incompressible magnetohydrodynamics equations. The rotational pressurecorrection (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 firstorder RPCSAVEuler

Efficient function approximation in enriched approximation spaces IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240512
Astrid Herremans, Daan HuybrechsAn 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 illconditioned system. Recent research indicates that these systems can

Error analysis for local discontinuous Galerkin semidiscretization of Richards’ equation IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240511
Scott Congreve, Vít Dolejší, Sunčica SakićThis 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 fastdiffusion type of degeneracy. We discretize the Richards’ equation by the local discontinuous

Variational data assimilation with finiteelement discretization for secondorder parabolic interface equation IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240511
Xuejian Li, Xiaoming He, Wei Gong, Craig C DouglasIn this paper, we propose and analyze a finiteelement method of variational data assimilation for a secondorder parabolic interface equation on a twodimensional 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

A linearly implicit finite element fulldiscretization scheme for SPDEs with nonglobally Lipschitz coefficients IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240508
Mengchao Wang, Xiaojie WangThe 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 wellposedness and regularity of the considered SPDEs in space dimension $d \le 3$, under more relaxed assumptions on the

Stability of convergence rates: kernel interpolation on nonLipschitz domains IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240508
Tizian Wenzel, Gabriele Santin, Bernard HaasdonkError 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

A certified waveletbased physicsinformed neural network for the solution of parameterized partial differential equations IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240505
Lewin Ernst, Karsten UrbanPhysics 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

Strong convergence of adaptive timestepping schemes for the stochastic Allen–Cahn equation IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240505
Chuchu Chen, Tonghe Dang, Jialin HongIt is known from Beccari et al. (2019) that the standard explicit Eulertype scheme (such as the exponential Euler and the linearimplicit 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 timestepping schemes, which adapt the timestep at each iteration

Finite element methods for multicomponent convectiondiffusion IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240428
Francis R A Aznaran, Patrick E Farrell, Charles W Monroe, Alexander J VanBruntWe develop finite element methods for coupling the steadystate 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

An explicit spectral Fletcher–Reeves conjugate gradient method for bicriteria optimization IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240412
Y Elboulqe, M El MaghriIn this paper, we propose a spectral Fletcher–Reeves conjugate gradientlike method for solving unconstrained bicriteria 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

On the rate of convergence of Yosida approximation for the nonlocal Cahn–Hilliard equation IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240410
Piotr Gwiazda, Jakub Skrzeczkowski, Lara TrussardiIt is wellknown 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

Monolithic and local timestepping decoupled algorithms for transport problems in fractured porous media IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240403
Yanzhao Cao, ThiThaoPhuong Hoang, PhuocToan HuynhThe objective of this paper is to develop efficient numerical algorithms for the linear advectiondiffusion 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 upwindmixed

On the necessity of the infsup condition for a mixed finite element formulation IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240228
Fleurianne Bertrand, Daniele BoffiWe 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 infsup

Pressure and convection robust bounds for continuous interior penalty divergencefree finite element methods for the incompressible Navier–Stokes equations IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240207
Bosco GarcíaArchilla, Julia NovoIn this paper, we analyze a pressurerobust method based on divergencefree 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

An Eulerian finite element method for the linearized Navier–Stokes problem in an evolving domain IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240130
Michael Neilan, Maxim OlshanskiiThe paper addresses an error analysis of an Eulerian finite element method used for solving a linearized Navier–Stokes problem in a timedependent 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 formulatype timestepping procedure with

Goaloriented error estimation based on equilibrated flux reconstruction for the approximation of the harmonic formulations in eddy current problems IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240129
Emmanuel Creusé, Serge Nicaise, Zuqi TangIn this work, we propose an a posteriori goaloriented 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

Cauchy data for Levin’s method IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240125
Anthony AshtonIn 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

Full operator preconditioning and the accuracy of solving linear systems IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240125
Stephan Mohr, Yuji Nakatsukasa, Carolina UrzúaTorresUnless special conditions apply, the attempt to solve illconditioned 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 illconditioning is tackled by means of preconditioning

Wellposedness and error estimates for coupled systems of nonlocal conservation laws IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240120
Aekta Aggarwal, Helge Holden, Ganesh VaidyaThis 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

Corrigendum to: Adaptive FEM with quasioptimal overall cost for nonsymmetric linear elliptic PDEs IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240119
Maximilian Brunner, Michael Innerberger, Ani Miraçi, Dirk Praetorius, Julian Streitberger, Pascal HeidUnfortunately, 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.

Convergence with rates for a Riccatibased discretization of SLQ problems with SPDEs IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240119
Andreas Prohl, Yanqing WangWe 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

Gammaconvergent LDG method for large bending deformations of bilayer plates IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240119
Andrea Bonito, Ricardo H Nochetto, Shuo YangBilayer 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

Darcy’s problem coupled with the heat equation under singular forcing: analysis and discretization IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240111
Alejandro Allendes, Gilberto Campaña, Francisco Fuica, Enrique OtárolaWe study the existence of solutions for Darcy’s problem coupled with the heat equation under singular forcing; the righthand 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

Compactness estimates for difference schemes for conservation laws with discontinuous flux IMA J. Numer. Anal. (IF 2.3) Pub Date : 20240104
Kenneth H Karlsen, John D TowersWe 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

A constraint dissolving approach for nonsmooth optimization over the Stiefel manifold IMA J. Numer. Anal. (IF 2.3) Pub Date : 20231223
Xiaoyin Hu, Nachuan Xiao, Xin Liu, KimChuan TohThis 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 proxfriendly objective functions. We propose a constraint dissolving function named NCDF and show that it has the same firstorder stationary points and local minimizers as the original problem in a neighborhood of the Stiefel

On best pnorm approximation of discrete data by polynomials IMA J. Numer. Anal. (IF 2.3) Pub Date : 20231209
Michael S FloaterIn 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.

Semilinear optimal control with Dirac measures IMA J. Numer. Anal. (IF 2.3) Pub Date : 20231129
Enrique OtárolaThe 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, secondorder optimality conditions. We develop

Unconditionally stable small stencil enriched multiple point flux approximations of heterogeneous diffusion problems on general meshes IMA J. Numer. Anal. (IF 2.3) Pub Date : 20231125
Julien CoatlévenWe 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 welldesigned stabilization term from which

Adaptive VEM for variable data: convergence and optimality IMA J. Numer. Anal. (IF 2.3) Pub Date : 20231116
L Beirão da Veiga, C Canuto, R H Nochetto, G Vacca, M VeraniWe 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 stabilizationfree a posteriori error estimators recently derived in Beirão da Veiga et al. (2023, Adaptive VEM: stabilizationfree a posteriori error analysis and contraction property. SIAM J. Numer. Anal., 61, 457–494). The crucial

Convergent finite element methods for the perfect conductivity problem with closetotouching inclusions IMA J. Numer. Anal. (IF 2.3) Pub Date : 20231114
Buyang Li, Haigang Li, Zongze YangIn 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 closetotouching inclusions of general

A twodimensional boundary value problem of elliptic type with nonlocal conjugation conditions IMA J. Numer. Anal. (IF 2.3) Pub Date : 20231114
Zorica Milovanović Jeknić, Aleksandra Delić, Sandra ŽivanovićWe consider an elliptic boundary value problem with nonlocal conjugation conditions. An a priori estimate for its weak solution in an appropriate Sobolevlike 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.

Numerical analysis of a hybridized discontinuous Galerkin method for the Cahn–Hilliard problem IMA J. Numer. Anal. (IF 2.3) Pub Date : 20231111
Keegan L A Kirk, Beatrice Riviere, Rami MasriThe 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

Twoscale methods for the normalized infinity Laplacian: rates of convergence IMA J. Numer. Anal. (IF 2.3) Pub Date : 20231111
Wenbo Li, Abner J SalgadoWe 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 socalled twoscale 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 righthand side vanishes

Weak error analysis for strong approximation schemes of SDEs with superlinear coefficients IMA J. Numer. Anal. (IF 2.3) Pub Date : 20231111
Xiaojie Wang, Yuying Zhao, Zhongqiang ZhangWe present an error analysis of weak convergence of onestep numerical schemes for stochastic differential equations (SDEs) with superlinearly growing coefficients. Following Milstein’s weak error analysis on the onestep approximation of SDEs, we prove a general result on weak convergence of the onestep discretization of the SDEs mentioned above. As applications, we show the weak convergence rates

Uniform L∞bounds for energyconserving higherorder time integrators for the Gross–Pitaevskii equation with rotation IMA J. Numer. Anal. (IF 2.3) Pub Date : 20231107
Christian Döding, Patrick HenningIn this paper, we consider an energyconserving 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)

Convergent evolving finite element approximations of boundary evolution under shape gradient flow IMA J. Numer. Anal. (IF 2.3) Pub Date : 20231030
Wei Gong, Buyang Li, Qiqi RaoAs 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 solutiondriven evolving boundary. Rigorous analysis for the stability and convergence of such finite element