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A twostage approximation strategy for piecewise smooth functions in two and three dimensions IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210923
Sergio Amat, David Levin, Juan RuizÁlvarezGiven values of a piecewise smooth function $f$ on a square grid within a domain $[0,1]^d$, $d=2,3$, we look for a piecewise adaptive approximation to $f$. Standard approximation techniques achieve reduced approximation orders near the boundary of the domain and near curves of jump singularities of the function or its derivatives. The insight used here is that the behavior near the boundaries

An Lp spacesbased formulation yielding a new fully mixed finite element method for the coupled Darcy and heat equations IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210910
Gatica G, Meddahi S, RuizBaier R.AbstractIn this work we present and analyse a new fully mixed finite element method for the nonlinear problem given by the coupling of the Darcy and heat equations. Besides the velocity, pressure and temperature variables of the fluid, our approach is based on the introduction of the pseudoheat flux as a further unknown. As a consequence of it, and due to the convective term involving the velocity

A leastsquares Galerkin approach to gradient and Hessian recovery for nondivergenceform elliptic equations IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210909
Lakkis O, Mousavi A.AbstractWe propose a leastsquares method involving the recovery of the gradient and possibly the Hessian for elliptic equation in nondivergence form. As our approach is based on the Lax–Milgram theorem with the curlfree constraint built into the target (or cost) functional, the discrete spaces require no infsup stabilization. We show that standard conforming finite elements can be used yielding

Finite element approximation of fractional Neumann problems IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210907
Francisco M Bersetche, Juan Pablo BorthagarayIn this paper, we consider approximations of Neumann problems for the integral fractional Laplacian by continuous, piecewise linear finite elements. We analyze the weak formulation of such problems, including their wellposedness and asymptotic behavior of solutions. We address the convergence of the finite element discretizations and discuss the implementation of the method. Finally, we present several

Shanks and Andersontype acceleration techniques for systems of nonlinear equations IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210825
Brezinski C, Cipolla S, RedivoZaglia M, et al.AbstractThis paper examines a number of extrapolation and acceleration methods and introduces a few modifications of the standard Shanks transformation that deal with general sequences. One of the goals of the paper is to lay out a general framework that encompasses most of the known acceleration strategies. The paper also considers the Anderson Acceleration (AA) method under a new light and exploits

The Kirchhoff plate equation on surfaces: the surface Hellan–Herrmann–Johnson method IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210820
Walker S.AbstractWe present a mixed finite element method for approximating a fourthorder elliptic partial differential equation (PDE), the Kirchhoff plate equation, on a surface embedded in ${\mathbb {R}}^{3}$, with or without boundary. Error estimates are given in meshdependent norms that account for the surface approximation and the approximation of the surface PDE. The method is built on the classic

An HDG method for dissimilar meshes IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210813
Manuel Solano, Sébastien Terrana, NgocCuong Nguyen, Jaime PeraireWe present a hybridizable discontinuous Galerkin (HDG) method for dissimilar meshes. The method is devised by formulating HDG discretizations on separate meshes and gluing these HDG discretizations through appropriate transmission conditions that weakly enforce the continuity of the numerical trace and the numerical flux across the dissimilar interfaces. The transmission conditions are based upon transferring

A highorder fully discrete scheme for the Korteweg–de Vries equation with a timestepping procedure of Runge–Kuttacomposition type IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210812
Vassilios A Dougalis, Ángel DuránWe consider the periodic initialvalue problem for the Korteweg–de Vries equation that we discretize in space by a spectral Fourier–Galerkin method and in time by an implicit, highorder, Runge–Kutta scheme of composition type based on the implicit midpoint rule. We prove $L^{2}$ error estimates for the resulting semidiscrete and the fully discrete approximations. Some numerical experiments illustrate

Uniform error estimates for artificial neural network approximations for heat equations IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210810
Lukas Gonon, Philipp Grohs, Arnulf Jentzen, David Kofler, David ŠiškaRecently, artificial neural networks (ANNs) in conjunction with stochastic gradient descent optimization methods have been employed to approximately compute solutions of possibly rather highdimensional partial differential equations (PDEs). Very recently, there have also been a number of rigorous mathematical results in the scientific literature, which examine the approximation capabilities of such

On the convergence and meshindependent property of the Barzilai–Borwein method for PDEconstrained optimization IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210805
Behzad Azmi, Karl KunischAiming at optimization problems governed by partial differential equations (PDEs), local Rlinear convergence of the Barzilai–Borwein (BB) method for a class of twice continuously Fréchetdifferentiable functions is proven. Relying on this result, the meshindependent principle for the BBmethod is investigated. The applicability of the theoretical results is demonstrated for two different types of

A lockingfree P0 finite element method for linear elasticity equations on polytopal partitions IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210805
Yujie Liu, Junping WangThis article presents a $P_0$ finite element method for boundary value problems for linear elasticity equations. The new method makes use of piecewise constant approximating functions on the boundary of each polytopal element and is devised by simplifying and modifying the weak Galerkin finite element method based on $P_1/P_0$ approximations for the displacement. This new scheme includes a tangential

Convergence of a multidimensional Glimmlike scheme for the transport of fronts IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210727
Thierry Gallouët, Olivier HurisseThis paper is devoted to the numerical analysis of a numerical scheme for the simulation of front advection. The scheme has been recently proposed and it is based on the ideas used for Glimm’s scheme. It relies on a twostep approach: a convection step is followed by a projection step, which is based on a random choice. The main advantage of this scheme is that it is applicable to multidimensional

Homogeneous multigrid for HDG IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210719
Lu P, Rupp A, Kanschat G.AbstractWe introduce a multigrid method that is homogeneous in the sense that it uses the same hybridizable discontinuous Galerkin (HDG) discretization scheme for Poisson’s equation on all levels. In particular, we construct a stable injection operator and prove optimal convergence of the method under the assumption of elliptic regularity. Numerical experiments underline our analytical findings.

Optimal convergence of a secondorder lowregularity integrator for the KdV equation IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210716
Yifei Wu, Xiaofei ZhaoIn this paper, we establish the optimal convergence for a secondorder exponentialtype integrator from Hofmanová & Schratz (2017, An exponentialtype integrator for the KdV equation. Numer. Math., 136, 1117–1137) for solving the Korteweg–de Vries equation with rough initial data. The scheme is explicit and efficient to implement. By rigorous error analysis, we show that the scheme provides secondorder

John W. Barrett 29 June 1955, Wimbledon – 30 June 2019, Wimbledon IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210625

Uniform Höldernorm bounds for finite element approximations of secondorder elliptic equations IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210512
Diening L, Scharle T, Süli E.AbstractWe develop a discrete counterpart of the De Giorgi–Nash–Moser theory, which provides uniform Höldernorm bounds on continuous piecewise affine finite element approximations of secondorder linear elliptic problems of the form $\nabla \cdot (A\nabla u)=f\nabla \cdot F$ with $A\in L^\infty (\varOmega ; {{\mathbb{R}}}^{n\times n})$ a uniformly elliptic matrixvalued function, $f\in L^{q}(\varOmega

An adaptive edge finite element DtN method for Maxwell’s equations in biperiodic structures IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210713
Xue Jiang, Peijun Li, Junliang Lv, Zhoufeng Wang, Haijun Wu, Weiying ZhengWe consider the diffraction of an electromagnetic plane wave by a biperiodic structure. This paper is concerned with a numerical solution of the diffraction grating problem for threedimensional Maxwell’s equations. Based on the DirichlettoNeumann (DtN) operator, an equivalent boundary value problem is formulated in a bounded domain by using a transparent boundary condition. An a posteriori error

Finitevolume approximation of the invariant measure of a viscous stochastic scalar conservation law IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210713
Sébastien Boyaval, Sofiane Martel, Julien ReygnerWe study the numerical approximation of the invariant measure of a viscous scalar conservation law, onedimensional and periodic in the space variable and stochastically forced with a whiteintime but spatially correlated noise. The flux function is assumed to be locally Lipschitz continuous and to have at most polynomial growth. The numerical scheme we employ discretizes the stochastic partial differential

Stable gradient flow discretizations for simulating bilayer plate bending with isometry and obstacle constraints IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210710
Sören Bartels, Christian PalusBilayer plates are compound materials that exhibit large bending deformations when exposed to environmental changes that lead to different mechanical responses in the involved materials. In this article a new numerical method that is suitable for simulating the isometric deformation induced by a given material mismatch in a bilayer plate is discussed. A dimensionally reduced formulation of the bending

An unfitted Eulerian finite element method for the timedependent Stokes problem on moving domains IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210705
von Wahl H, Richter T, Lehrenfeld C.AbstractWe analyse a Eulerian finite element method, combining a Eulerian timestepping scheme applied to the timedependent Stokes equations with the CutFEM approach using infsup stable Taylor–Hood elements for the spatial discretization. This is based on the method introduced by Lehrenfeld & Olshanskii (2019, A Eulerian finite element method for PDEs in timedependent domains. ESAIM: M2AN, 53, 585–614)

Consistency analysis of the Grünwald–Letnikov approximation in a bounded domain IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210702
Sousa E.AbstractThe Grünwald–Letnikov approximation is a wellknown discretization to approximate a Riemann–Liouville derivative of order $\alpha>0$. This approximation has been proved to be a consistent approximation, of order $1 $, when the domain is the real line, using Fourier transforms. However, in recent years, this approximation has been applied frequently to solve fractional differential equations

Numerical methods for stochastic Volterra integral equations with weakly singular kernels IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210629
Min Li, Chengming Huang, Yaozhong HuIn this paper we first establish the existence, uniqueness and Hölder continuity of the solution to stochastic Volterra integral equations (SVIEs) with weakly singular kernels, with singularities $\alpha \in (0, 1)$ for the drift term and $\beta \in (0, 1/2)$ for the stochastic term. Subsequently, we propose a $\theta $Euler–Maruyama scheme and a Milstein scheme to solve the equations numerically

Error estimates for the Cahn–Hilliard equation with dynamic boundary conditions IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210623
Paula Harder, Balázs KovácsA proof of convergence is given for a bulk–surface finite element semidiscretisation of the Cahn–Hilliard equation with Cahn–Hilliardtype dynamic boundary conditions in a smooth domain. The semidiscretisation is studied in an abstract weak formulation as a secondorder system. Optimalorder uniformintime error estimates are shown in the $L^2$ and $H^1$norms. The error estimates are based on

Error estimate of a decoupled numerical scheme for the Cahn–Hilliard–Stokes–Darcy system IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210623
Chen W, Wang S, Zhang Y, et al.AbstractWe analyze a fully discrete finite element numerical scheme for the Cahn–Hilliard–Stokes–Darcy system that models twophase flows in coupled free flow and porous media. To avoid a wellknown difficulty associated with the coupling between the Cahn–Hilliard equation and the fluid motion, we make use of the operatorsplitting in the numerical scheme, so that these two solvers are decoupled, which

A convergent finite element algorithm for generalized mean curvature flows of closed surfaces IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210623
Binz T, Kovács B.AbstractAn algorithm is proposed for generalized mean curvature flow of closed twodimensional surfaces, which include inverse mean curvature flow and powers of mean and inverse mean curvature flow. Error estimates are proved for semidiscretizations and full discretizations for the generalized flow. The algorithm proposed and studied here combines evolving surface finite elements, whose nodes determine

Transfer of conservative discrete differential operators between staggered grids: construction and duality relations IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210623
Goudon T, Krell S, Llobell J, et al.AbstractThis paper is concerned with the construction of a discrete conservative operator on a mesh, knowing a conservative operator on a different mesh. We analyse the possibility of defining such a transfer based on the preservation of conservation properties and provide practical procedures, which can be effectively implemented for schemes that work on staggered grids. We show that such a transfer

Estimates on the generalization error of physicsinformed neural networks for approximating a class of inverse problems for PDEs IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210617
Mishra S, Molinaro R.AbstractPhysicsinformed neural networks (PINNs) have recently been very successfully applied for efficiently approximating inverse problems for partial differential equations (PDEs). We focus on a particular class of inverse problems, the socalled data assimilation or unique continuation problems, and prove rigorous estimates on the generalization error of PINNs approximating them. An abstract framework

Local multigrid solvers for adaptive isogeometric analysis in hierarchical spline spaces IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210615
Clemens Hofreither, Ludwig Mitter, Hendrik SpeleersWe propose local multigrid solvers for adaptively refined isogeometric discretizations using (truncated) hierarchical Bsplines ((T)HBsplines). Smoothing is only performed in or near the refinement areas on each level, leading to a computationally efficient solving strategy. We prove robust convergence of the proposed solvers with respect to the number of levels and the mesh sizes of the hierarchical

The transport of images method: computing all zeros of harmonic mappings by continuation IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210609
Olivier Sète, Jan ZurWe present a continuation method to compute all zeros of a harmonic mapping $\,f$ in the complex plane. Our method works without any prior knowledge of the number of zeros or their approximate location. We start by computing all solutions of $f(z) = \eta $ with $\lvert \eta \rvert{}$ sufficiently large and then track all solutions as $\eta $ tends to $0$ to finally obtain all zeros of $f$

Galerkin finite element approximation of a stochastic semilinear fractional subdiffusion with fractionally integrated additive noise IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210521
Kang W, Egwu B, Yan Y, et al.AbstractA Galerkin finite element method is applied to approximate the solution of a semilinear stochastic space and time fractional subdiffusion problem with the Caputo fractional derivative of the order $ \alpha \in (0, 1)$, driven by fractionally integrated additive noise. After discussing the existence, uniqueness and regularity results, we approximate the noise with the piecewise constant function

A linear implicit Euler method for the finite element discretization of a controlled stochastic heat equation IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210520
Benner P, Stillfjord T, Trautwein C.AbstractWe consider a numerical approximation of a linear quadratic control problem constrained by the stochastic heat equation with nonhomogeneous Neumann boundary conditions. This involves a combination of distributed and boundary control, as well as both distributed and boundary noise. We apply the finite element method for the spatial discretization and the linear implicit Euler method for the

Convergence and rate optimality of adaptive multilevel stochastic Galerkin FEM IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210519
Bespalov A, Praetorius D, Ruggeri M.AbstractWe analyze an adaptive algorithm for the numerical solution of parametric elliptic partial differential equations in twodimensional physical domains, with coefficients and righthandside functions depending on infinitely many (stochastic) parameters. The algorithm generates multilevel stochastic Galerkin approximations; these are represented in terms of a sparse generalized polynomial chaos

The timefractional Cahn–Hilliard equation: analysis and approximation IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210519
AlMaskari M, Karaa S.AbstractWe consider a timefractional Cahn–Hilliard equation where the conventional firstorder time derivative is replaced by a Caputo fractional derivative of order $\alpha \in (0,1)$. Based on an a priori bound of the exact solution, global existence of solutions is proved and detailed regularity results are included. A finite element method is then analyzed in a spatially discrete case and in a

Analysis of backward Euler projection FEM for the Landau–Lifshitz equation IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210519
Rong An, Weiwei SunThe paper focuses on the analysis of the Euler projection Galerkin finite element method (FEM) for the dynamics of magnetization in ferromagnetic materials, described by the Landau–Lifshitz equation with the pointwise constraint ${\textbf{m}}=1$. The method is based on a simple sphere projection that projects the numerical solution onto a unit sphere at each time step, and the method has been

The Scharfetter–Gummel scheme for aggregation–diffusion equations IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210518
André Schlichting, Christian SeisIn this paper we propose a finitevolume scheme for aggregation–diffusion equations based on a Scharfetter–Gummel approximation of the quadratic, nonlocal flux term. This scheme is analyzed concerning well posedness and convergence towards solutions to the continuous problem. Also, it is proven that the numerical scheme has several structurepreserving features. More specifically, it is shown that

QuasiNewton variable preconditioning for nonlinear nonuniformly monotone elliptic problems posed in Banach spaces IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210518
B Borsos, J KarátsonQuasiNewtontype iterative solvers are developed for a wide class of nonlinear elliptic problems. The presented generalization of earlier efficient methods covers various nonuniformly elliptic problems arising, e.g., in nonNewtonian flows or for certain glaciology models. The robust estimates are reinforced by several examples.

Deep neural network approximation for highdimensional elliptic PDEs with boundary conditions IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210510
Philipp Grohs, Lukas HerrmannIn recent work it has been established that deep neural networks (DNNs) are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been restricted to problems formulated on the whole Euclidean domain. On the other hand, most problems in engineering and in the sciences are formulated on finite

Numerical analysis of a wave equation for lossy media obeying a frequency power law IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210510
Katherine Baker, Lehel BanjaiWe study a wave equation with a nonlocal time fractional damping term that models the effects of acoustic attenuation characterized by a frequencydependent power law. First we prove the existence of a unique solution to this equation with particular attention paid to the handling of the fractional derivative. Then we derive an explicit timestepping scheme based on the finite element method in space

Finite element analysis for a diffusion equation on a harmonically evolving domain IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210505
Dominik EdelmannWe study convergence of the evolving finite element semidiscretization of a parabolic partial differential equation on an evolving bulk domain. The boundary of the domain evolves with a given velocity, which is then extended to the bulk by solving a Poisson equation. The numerical solution to the parabolic equation depends on the numerical evolution of the bulk, which yields the timedependent mesh

Stability analysis of general multistep methods for Markovian backward stochastic differential equations IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210505
Xiao Tang, Jie XiongThis paper focuses on the stability analysis of a general class of linear multistep methods for decoupled forward–backward stochastic differential equations (FBSDEs). The general linear multistep methods we consider contain many wellknown linear multistep methods from the ordinary differential equation framework, such as Adams, Nyström, MilneSimpson and backward differentiation formula methods.

Strictly positive definite kernels on the 2sphere: from radial symmetry to eigenvalue block structure IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210503
Martin Buhmann, Janin JägerThe paper introduces a new characterization of strict positive definiteness for kernels on the 2sphere without assuming the kernel to be radially (isotropic) or axially symmetric. The results use the series expansion of the kernel in spherical harmonics. Then additional sufficient conditions are proven for kernels with a block structure of expansion coefficients. These generalize the result derived

Derivativefree highorder uniformly accurate schemes for highly oscillatory systems IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210503
Philippe Chartier, Mohammed Lemou, Florian Méhats, Xiaofei ZhaoIn this paper we address the computational aspects of uniformly accurate numerical methods for solving highly oscillatory evolution equations. In particular, we introduce an approximation strategy that allows the construction of arbitrary highorder methods using solely the righthand side of the differential equation. No derivative of the vector field is required, while uniform accuracy is retained

On the convergence of augmented Lagrangian strategies for nonlinear programming IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210427
Roberto Andreani, Ariel R Velazco, Alberto Ramos, Ademir A Ribeiro, Leonardo D SecchinAugmented Lagrangian (AL) algorithms are very popular and successful methods for solving constrained optimization problems. Recently, global convergence analysis of these methods has been dramatically improved by using the notion of sequential optimality conditions. Such conditions are necessary for optimality, regardless of the fulfillment of any constraint qualifications, and provide theoretical

On the rate of convergence of the Gaver–Stehfest algorithm IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210421
Alexey Kuznetsov, Justin MilesThe Gaver–Stehfest algorithm is widely used for numerical inversion of the Laplace transform. In this paper we provide the first rigorous study of the rate of convergence of the Gaver–Stehfest algorithm. We prove that Gaver–Stehfest approximations converge exponentially fast if the target function is analytic in a neighbourhood of a point and they converge at a rate $o(n^{k})$ if the target function

Maximal regularity of multistep fully discrete finite element methods for parabolic equations IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210409
Buyang LiThis article extends the semidiscrete maximal $L^p$regularity results in Li (2019, Analyticity, maximal regularity and maximumnorm stability of semidiscrete finite element solutions of parabolic equations in nonconvex polyhedra. Math. Comp., 88, 144) to multistep fully discrete finite element methods for parabolic equations with more general diffusion coefficients in $W^{1,d+\beta }$, where

A posteriori error estimation and spacetime adaptivity for a linear stochastic PDE with additive noise IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210325
Ananta K Majee, Andreas ProhlWe present a strong residualbased a posteriori error estimate for a finite elementbased spacetime discretization of the linear stochastic convected heat equation with additive noise. This error estimate is used for an adaptive algorithm that automatically selects deterministic mesh parameters in space and time. For every $n \geq 0$, we find a new timestep $\tau _n$, a new spatial mesh ${\mathcal

Strong convergence of a halfexplicit Euler scheme for constrained stochastic mechanical systems IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210324
Felix Lindner, Holger StrootThis paper is concerned with the numerical approximation of stochastic mechanical systems with nonlinear holonomic constraints. The considered systems are described by secondorder stochastic differentialalgebraic equations involving an implicitly given Lagrange multiplier process. The explicit representation of the Lagrange multiplier leads to an underlying stochastic ordinary differential equation

Optimally convergent mixed finite element methods for the stochastic Stokes equations IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210317
Feng X, Prohl A, Vo L.AbstractWe propose some new mixed finite element methods for the timedependent stochastic Stokes equations with multiplicative noise, which use the Helmholtz decomposition of the driving multiplicative noise. It is known (Langa, J. A., Real, J. & Simon, J. (2003) Existence and regularity of the pressure for the stochastic NavierStokes equations. Appl. Math. Optim., 48, 195210) that the pressure

Equivalence of local and globalbest approximations, a simple stable local commuting projector, and optimal hp approximation estimates in H(div) IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210316
Alexandre Ern, Thirupathi Gudi, Iain Smears, Martin VohralíkGiven an arbitrary function in $\boldsymbol{H}({\operatorname{div}})$, we show that the error attained by the globalbest approximation by $\boldsymbol{H}({\operatorname{div}})$conforming piecewise polynomial Raviart–Thomas–Nédélec elements under additional constraints on the divergence and normal flux on the boundary is, up to a generic constant, equivalent to the sum of independent localbest

The numerical unified transform method for initialboundary value problems on the halfline IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210304
Bernard Deconinck, Thomas Trogdon, Xin YangWe implement the unified transform method of Fokas as a numerical method to solve linear evolution partial differential equations on the halfline. The method computes the solution at any $x$ and $t$ without spatial discretization or time stepping. With the help of contour deformations and oscillatory integration techniques, the method’s complexity does not increase for large $x,t$ and the method

Plain convergence of adaptive algorithms without exploiting reliability and efficiency IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210304
Gregor Gantner, Dirk PraetoriusWe consider $h$adaptive algorithms in the context of the finite element method and the boundary element method. Under quite general assumptions on the building blocks SOLVE, ESTIMATE, MARK and REFINE of such algorithms we prove plain convergence in the sense that the adaptive algorithm drives the underlying a posteriori error estimator to zero. Unlike available results in the literature, our analysis

First order leastsquares formulations for eigenvalue problems IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210304
Fleurianne Bertrand, Daniele BoffiIn this paper we discuss spectral properties of operators associated with the leastsquares finiteelement approximation of elliptic partial differential equations. The convergence of the discrete eigenvalues and eigenfunctions towards the corresponding continuous eigenmodes is studied and analyzed with the help of appropriate $L^2$ error estimates. A priori and a posteriori estimates are proved

Design and convergence analysis of numerical methods for stochastic evolution equations with Leray–Lions operator IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210304
Jérôme Droniou, Beniamin Goldys, KimNgan LeThe gradient discretization method (GDM) is a generic framework, covering many classical methods (finite elements, finite volumes, discontinuous Galerkin, etc.), for designing and analysing numerical schemes for diffusion models. In this paper we study the GDM for a general stochastic evolution problem based on a Leray–Lions type operator. The problem contains the stochastic $p$Laplace equation

Numerical homogenization for nonlinear strongly monotone problems IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210225
Barbara VerfürthIn this work we introduce and analyse a new multiscale method for strongly nonlinear monotone equations in the spirit of the localized orthogonal decomposition. A problemadapted multiscale space is constructed by solving linear local finescale problems, which is then used in a generalized finite element method. The linearity of the finescale problems allows their localization and, moreover, makes

A family of fast fixed point iterations for M/G/1type Markov chains IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210225
Dario A Bini, Guy Latouche, Beatrice MeiniWe consider the problem of computing the minimal nonnegative solution $G$ of the nonlinear matrix equation $X=\sum _{i=1}^\infty A_iX^{i+1}$ where $A_i$, for $i\geqslant 1$, are nonnegative square matrices such that $\sum _{i=1}^\infty A_i$ is stochastic. This equation is fundamental in the analysis of M/G/1type Markov chains, since the matrix $G$ provides probabilistic measures

Simulation of constrained elastic curves and application to a conical sheet indentation problem IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210224
Bartels S.AbstractWe consider variational problems that model the bending behavior of curves that are constrained to belong to given hypersurfaces. Finite element discretizations of corresponding functionals are justified rigorously via $\varGamma $convergence. The stability of semiimplicit discretizations of gradient flows is investigated, which provide a practical method to determine stationary configurations

A variational approach to the sum splitting scheme IMA J. Numer. Anal. (IF 2.601) Pub Date : 20210121
Monika Eisenmann, Eskil HansenNonlinear parabolic equations are frequently encountered in applications and efficient approximating techniques for their solution are of great importance. In order to provide an effective scheme for the temporal approximation of such equations, we present a sum splitting scheme that comes with a straightforward parallelization strategy. The convergence analysis is carried out in a variational framework

Convergence analysis of subdivision processes on the sphere IMA J. Numer. Anal. (IF 2.601) Pub Date : 20201223
Svenja Hüning, Johannes WallnerWe analyse the convergence of nonlinear Riemannian analogues of linear subdivision processes operating on data in the sphere. We show how for curve subdivision rules we can derive bounds guaranteeing convergence if the density of input data is below that threshold. Previous results only yield thresholds that are several magnitudes smaller and are thus useless for a priori checking of convergence. It

Variational convergence of discrete elasticae IMA J. Numer. Anal. (IF 2.601) Pub Date : 20201222
Sebastian Scholtes, Henrik Schumacher, Max WardetzkyWe discuss a discretization of the Euler–Bernoulli bending energy and of Euler elasticae under clamped boundary conditions by polygonal lines. We show Hausdorff convergence of the set of almost minimizers of the discrete bending energy to the set of smooth Euler elasticae under mesh refinement in (i) the $W^{1,\infty }$topology for piecewiselinear interpolation; and in (ii) the $W^{2,p}$topology

Numerical analysis for nematic electrolytes IMA J. Numer. Anal. (IF 2.601) Pub Date : 20201212
Ľubomír Baňas, Robert Lasarzik, Andreas ProhlWe consider a system of nonlinear PDEs modeling nematic electrolytes, and construct a dissipative solution with the help of its implementable, structureinheriting and space–time discretization. Computational studies are performed to study the mutual effects of electric, elastic and viscous effects onto the molecules in a nematic electrolyte.