• A finite element method for degenerate two-phase flow in porous media. Part I: Well-posedness
J. Numer. Math. (IF 3.778) Pub Date : 2021-06-01
Vivette Girault, Beatrice Riviere, Loic Cappanera

A finite element method with mass-lumping and flux upwinding is formulated for solving the immiscible two-phase flow problem in porous media. The method approximates directly thewetting phase pressure and saturation, which are the primary unknowns. The discrete saturation satisfies a maximum principle. Stability of the scheme and existence of a solution are established.

• Numerical analysis of a stable discontinuous Galerkin scheme for the hydrostatic Stokes problem
J. Numer. Math. (IF 3.778) Pub Date : 2021-06-01
Francisco Guillén-Gonzàlez, M. Victoria Redondo-Neble, J. Rafael Rodríguez-Galvàn

We propose a Discontinuous Galerkin (DG) scheme for the hydrostatic Stokes equations. These equations, related to large-scale PDE models in oceanography, are characterized by the loss of ellipticity of the vertical momentum equation. This fact provides some interesting challenges, such as the design of stable numerical schemes. The new scheme proposed here is based on the symmetric interior penalty

• Frontmatter
J. Numer. Math. (IF 3.778) Pub Date : 2021-06-01

Article Frontmatter was published on June 1, 2021 in the journal Journal of Numerical Mathematics (volume 29, issue 2).

• Relative error analysis of matrix exponential approximations for numerical integration
J. Numer. Math. (IF 3.778) Pub Date : 2021-06-01
Stefano Maset

In this paper, we study the relative error in the numerical solution of a linear ordinary differential equation y '( t ) = Ay ( t ), t ≥ 0, where A is a normal matrix. The numerical solution is obtained by using at any step an approximation of the matrix exponential, e.g., a polynomial or a rational approximation. The error of the numerical solution with respect to the exact solution is due to this

• A redistributed bundle algorithm based on local convexification models for nonlinear nonsmooth DC programming
J. Numer. Math. (IF 3.778) Pub Date : 2021-06-01
Jie Shen, Jia-Tong Li, Fang-Fang Guo, Na Xu

For nonlinear nonsmooth DC programming (difference of convex functions), we introduce a new redistributed proximal bundle method. The subgradient information of both the DC components is gathered from some neighbourhood of the current stability center and it is used to build separately an approximation for each component in the DC representation. Especially we employ the nonlinear redistributed technique

• Frontmatter
J. Numer. Math. (IF 3.778) Pub Date : 2021-03-01

Article Frontmatter was published on March 1, 2021 in the journal Journal of Numerical Mathematics (volume 29, issue 1).

• Convergence of time-splitting approximations for degenerate convection–diffusion equations with a random source
J. Numer. Math. (IF 3.778) Pub Date : 2021-03-01

In this paper we propose a time-splitting method for degenerate convection–diffusion equations perturbed stochastically by white noise. This work generalizes previous results on splitting operator techniques for stochastic hyperbolic conservation laws for the degenerate parabolic case. The convergence in Llocp$\begin{array}{} \displaystyle L^p_{\rm loc} \end{array}$ of the time-splitting operator scheme

• A two-grid method with backtracking for the mixed Stokes/Darcy model
J. Numer. Math. (IF 3.778) Pub Date : 2021-03-01
Guangzhi Du, Liyun Zuo

In this paper, a two-grid method with backtracking is proposed and investigated for the mixed Stokes/Darcy system which describes a fluid flow coupled with a porous media flow. Based on the classical two-grid method , a coarse mesh correction is carried out to derive optimal error bounds for the velocity field and the piezometric head in L 2 norm. Finally, results of numerical experiments are provided

• Collocated finite-volume method for the incompressible Navier–Stokes problem
J. Numer. Math. (IF 3.778) Pub Date : 2021-03-01
Kirill M. Terekhov

A collocated finite-volume method for the incompressible Navier–Stokes problem is introduced. The method applies to general polyhedral grids and demonstrates higher than the first order of convergence. The velocity components and the pressure are approximated by piecewise-linear continuous and piecewise-constant fields, respectively. The method does not require artificial boundary conditions for pressure

• A finite element method for degenerate two-phase flow in porous media. Part II: Convergence
J. Numer. Math. (IF 3.778) Pub Date : 2021-01-16
Vivette Girault, Beatrice Riviere, Loic Cappanera

Convergence of a finite element method with mass-lumping and flux upwinding is formulated for solving the immiscible two-phase flow problem in porous media. The method approximates directly the wetting phase pressure and saturation, which are the primary unknowns. Well-posedness is obtained in []. Theoretical convergence is proved via a compactness argument. The numerical phase saturation converges

• Schur complement spectral bounds for large hybrid FETI-DP clusters and huge three-dimensional scalar problems
J. Numer. Math. (IF 3.778) Pub Date : 2021-01-16
Zdeněk Dostál, Tomáš Brzobohatý, Oldřich Vlach

Bounds on the spectrum of Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients of the convergence analysis of FETI (finite element tearing and interconnecting) based domain decomposition methods. Here we give bounds on the regular condition number of Schur complements of “floating” clusters arising from the discretization of 3D Laplacian on a

• P1-nonconforming divergence-free finite element method on square meshes for Stokes equations
J. Numer. Math. (IF 3.778) Pub Date : 2020-12-16
Chunjae Park

Abstract Recently, the P1-nonconforming finite element space over square meshes has been proved stable to solve Stokes equations with the piecewise constant space for velocity and pressure, respectively. In this paper, we will introduce its locally divergence-free subspace to solve the elliptic problem for the velocity only decoupled from the Stokes equation. The concerning system of linear equations

• Overcoming the curse of dimensionality in the numerical approximation of Allen–Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations
J. Numer. Math. (IF 3.778) Pub Date : 2020-12-16
Christian Beck, Fabian Hornung, Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse

Abstract One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite elements suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the dimension. In this work we overcome

• Coupling of virtual element and boundary element methods for the solution of acoustic scattering problems
J. Numer. Math. (IF 3.778) Pub Date : 2020-12-16
Gabriel N. Gatica, Salim Meddahi

Abstract This paper extends the applicability of the combined use of the virtual element method (VEM) and the boundary element method (BEM), recently introduced to solve the coupling of linear elliptic equations in divergence form with the Laplace equation, to the case of acoustic scattering problems in 2D and 3D. The well-posedness of the continuous and discrete formulations are established, and then

• Matrix equation solving of PDEs in polygonal domains using conformal mappings
J. Numer. Math. (IF 3.778) Pub Date : 2020-11-26
Yue Hao, Valeria Simoncini

We explore algebraic strategies for numerically solving linear elliptic partial differential equations in polygonal domains. To discretize the polygon by means of structured meshes, we employ Schwarz-Christoffel conformal mappings, leading to a multiterm linear equation possibly including Hadamard products of some of the terms. This new algebraic formulation allows us to clearly distinguish between

• Frontmatter
J. Numer. Math. (IF 3.778) Pub Date : 2020-12-01

Article Frontmatter was published on December 1, 2020 in the journal Journal of Numerical Mathematics (volume 28, issue 4).

• Acceleration of nonlinear solvers for natural convection problems
J. Numer. Math. (IF 3.778) Pub Date : 2020-11-14
Sara Pollock, Leo G. Rebholz, Mengying Xiao

This paper develops an efficient and robust solution technique for the steady Boussinesq model of non-isothermal flow using Anderson acceleration applied to a Picard iteration. After analyzing the fixed point operator associated with the nonlinear iteration to prove that certain stability and regularity properties hold, we apply the authors' recently constructed theory for Anderson acceleration, which

• Numerical simulation for European and American option of risks in climate change of Three Gorges Reservoir Area
J. Numer. Math. (IF 3.778) Pub Date : 2020-10-29
Fei Huang, Zuliang Lu, Lin Li, Xiankui Wu, Shang Liu, Yin Yang

With the climate change processes over times, all professions and trades in Three Gorges Reservoir Area will be influenced. One of the biggest challenges is the risk of rising sea level. In this situation, a large number of uncertainties for climate changes will be faced in Three Gorges Reservoir Area. Therefore, it is of importance to investigate the complexity of decision making on investing in the

• Entropy stabilization and property-preserving limiters for ℙ1 discontinuous Galerkin discretizations of scalar hyperbolic problems
J. Numer. Math. (IF 3.778) Pub Date : 2020-11-03
Dmitri Kuzmin

The methodology proposed in this paper bridges the gap between entropy stable and positivity-preserving discontinuous Galerkin (DG) methods for nonlinear hyperbolic problems. The entropy stability property and, optionally, preservation of local bounds for cell averages are enforced using flux limiters based on entropy conditions and discrete maximum principles, respectively. Entropy production by the

• Error analysis of higher order trace finite element methods for the surface Stokes equation
J. Numer. Math. (IF 3.778) Pub Date : 2020-10-04
Thomas Jankuhn, Maxim A. Olshanskii, Arnold Reusken, Alexander Zhiliakov

The paper studies a higher order unfitted finite element method for the Stokes system posed on a surface in three-dimensional space. The method employs generalized Taylor-Hood finite element pairs on tetrahedral bulk mesh to discretize the Stokes system on embedded surface. Stability and optimal order convergence results are proved. The proofs include a complete quantification of geometric errors stemming

• Doubly-Adaptive Artificial Compression Methods for Incompressible Flow
J. Numer. Math. (IF 3.778) Pub Date : 2020-09-25
William Layton, Michael McLaughlin

Abstract This report presents adaptive artificial compression methods in which the time-step and artificial compression parameter ε are independently adapted. The resulting algorithms are supported by analysis and numerical tests. The first and second-order methods are embedded. As a result, the computational, cognitive, and space complexities of the adaptive ε, k algorithms are negligibly greater

• Families of Interior Penalty Hybridizable discontinuous Galerkin methods for second order elliptic problems
J. Numer. Math. (IF 3.778) Pub Date : 2020-09-25
Maurice S. Fabien, Matthew G. Knepley, Beatrice M. Riviere

Abstract The focus of this paper is the analysis of families of hybridizable interior penalty discontinuous Galerkin methods for second order elliptic problems. We derive a priori error estimates in the energy norm that are optimal with respect to the mesh size. Suboptimal L2-norm error estimates are proven. These results are valid in two and three dimensions. Numerical results support our theoretical

• The deal.II Library, Version 9.2
J. Numer. Math. (IF 3.778) Pub Date : 2020-09-25
Daniel Arndt, Wolfgang Bangerth, Bruno Blais, Thomas C. Clevenger, Marc Fehling, Alexander V. Grayver, Timo Heister, Luca Heltai, Martin Kronbichler, Matthias Maier, Peter Munch, Jean-Paul Pelteret, Reza Rastak, Ignacio Tomas, Bruno Turcksin, Zhuoran Wang, David Wells

Abstract This paper provides an overview of the new features of the finite element library deal.II, version 9.2.

• Reduced basis approximations of the solutions to spectral fractional diffusion problems
J. Numer. Math. (IF 3.778) Pub Date : 2020-09-25
Andrea Bonito, Diane Guignard, Ashley R. Zhang

Abstract We consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction–diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced

• Frontmatter
J. Numer. Math. (IF 3.778) Pub Date : 2020-09-14

Journal Name: Journal of Numerical Mathematics Volume: 28 Issue: 3 Pages: i-iii

• Boundary update via resolvent for fluid-structure interaction
J. Numer. Math. (IF 3.778) Pub Date : 2020-06-30
Martina Bukač, Catalin Trenchea

We propose propose a BOundary Update using Resolvent (BOUR) partitioned method, second-order accurate in time, unconditionally stable, for the interaction between a viscous, incompressible fluid and a thin structure. The method is algorithmically similar to the sequential Backward Euler Forward Euler implementation of the midpoint quadrature rule. (i) The structure and fluid sub-problems are first

• A note on the efficient evaluation of a modified Hilbert transformation
J. Numer. Math. (IF 3.778) Pub Date : 2020-06-30
Olaf Steinbach, Marco Zank

In this note we consider an efficient data–sparse approximation of a modified Hilbert type transformation as it is used for the space–time finite element discretization of parabolic evolution equations in the anisotropic Sobolev space H1,1/2(Q). The resulting bilinear form of the first order time derivative is symmetric and positive definite, and similar as the integration by parts formula for the

• On convergent schemes for a two-phase Oldroyd-B type model with variable polymer density
J. Numer. Math. (IF 3.778) Pub Date : 2020-06-25
Oliver Sieber

Abstract The paper is concerned with a diffuse-interface model that describes two-phase flow of dilute polymeric solutions with a variable particle density. The additional stresses, which arise by elongations of the polymers caused by deformations of the fluid, are described by Kramers stress tensor. The evolution of Kramers stress tensor is modeled by an Oldroyd-B type equation that is coupled to

• The Fourier-finite-element method for Poisson’s equation in three-dimensional axisymmetric domains with edges: Computing the edge flux intensity functions
J. Numer. Math. (IF 3.778) Pub Date : 2020-06-25
Boniface Nkemzi, Michael Jung

Abstract In [Nkemzi and Jung, 2013] explicit extraction formulas for the computation of the edge flux intensity functions for the Laplacian at axisymmetric edges are presented. The present paper proposes a new adaptation for the Fourier-finite-element method for efficient numerical treatment of boundary value problems for the Poisson equation in axisymmetric domains Ω̂ ⊂ ℝ3 with edges. The novelty

• On generalized binomial laws to evaluate finite element accuracy: preliminary probabilistic results for adaptive mesh refinement
J. Numer. Math. (IF 3.778) Pub Date : 2020-06-25

Abstract The aim of this paper is to provide new perspectives on the relative finite elements accuracy. Starting from a geometrical interpretation of the error estimate which can be deduced from Bramble–Hilbert lemma, we derive a probability law that evaluates the relative accuracy, considered as a random variable, between two finite elements Pk and Pm, k < m. We extend this probability law to get

• Frontmatter
J. Numer. Math. (IF 3.778) Pub Date : 2020-06-11

Journal Name: Journal of Numerical Mathematics Volume: 28 Issue: 2 Pages: i-iii

• A decoupled finite element method with different time steps for the nonstationary Darcy-Brinkman problem
J. Numer. Math. (IF 3.778) Pub Date : 2020-03-26
Cheng Liao, Pengzhan Huang, Yinnian He

Abstract A decoupled finite element method with different time steps for the nonstationary Darcy--Brinkman problem is considered in this paper. Moreover, for the presented method, the stability analysis and error estimates are deduced. Finally, numerical tests are provided that demonstrate the efficiency of the method. It is found the presented method can save lots of computational time compared with

• Some transpose-free CG-like solvers for nonsymmetric ill-posed problems
J. Numer. Math. (IF 3.778) Pub Date : 2020-03-26
Silvia Gazzola, Paolo Novati

Abstract This paper introduces and analyzes an original class of Krylov subspace methods that provide an efficient alternative to many well-known conjugate-gradient-like (CG-like) Krylov solvers for square nonsymmetric linear systems arising from discretizations of inverse ill-posed problems. The main idea underlying the new methods is to consider some rank-deficient approximations of the transpose

• Inexact Newton method for the solution of eigenproblems arising in hydrodynamic temporal stability analysis
J. Numer. Math. (IF 3.778) Pub Date : 2020-03-26
Kirill V. Demyanko, Igor E. Kaporin, Yuri M. Nechepurenko

Abstract The inexact Newton method developed earlier for computing deflating subspaces associated with separated groups of finite eigenvalues of regular linear large sparse non-Hermitian matrix pencils is specialized to solve eigenproblems arising in the hydrodynamic temporal stability analysis. To this end, for linear systems to be solved at each step of the Newton method, a new efficient MLILU2 preconditioner

• Frontmatter
J. Numer. Math. (IF 3.778) Pub Date : 2020-03-26

Journal Name: Journal of Numerical Mathematics Volume: 28 Issue: 1 Pages: i-iii

• The deal.II Library, Version 9.1
J. Numer. Math. (IF 3.778) Pub Date : 2019-12-18
Daniel Arndt, Wolfgang Bangerth, Thomas C. Clevenger, Denis Davydov, Marc Fehling, Daniel Garcia-Sanchez, Graham Harper, Timo Heister, Luca Heltai, Martin Kronbichler, Ross Maguire Kynch, Matthias Maier, Jean-Paul Pelteret, Bruno Turcksin, David Wells

Abstract This paper provides an overview of the new features of the finite element library deal.II, version 9.1.

• Multigoal-Oriented Error Estimates for Non-linear Problems
J. Numer. Math. (IF 3.778) Pub Date : 2019-12-18
Bernhard Endtmayer, Ulrich Langer, Thomas Wick

Abstract In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the Partial Differential Equation (PDE) and the goal functionals may be nonlinear. Our method is based on a posteriori error estimates in which the adjoint problem

• A flux-corrected RBF-FD method for convection dominated problems in domains and on manifolds
J. Numer. Math. (IF 3.778) Pub Date : 2019-12-18
Andriy Sokolov, Oleg Davydov, Dmitri Kuzmin, Alexander Westermann, Stefan Turek

Abstract In this work, we present a Flux-Corrected Transport (FCT) algorithm for enforcing discrete maximum principles in Radial Basis Function (RBF) generalized Finite Difference (FD) methods for convection-dominated problems. The algorithm is constructed to guarantee mass conservation and to preserve positivity of the solution for irregular data nodes. The method can be applied both for problems

• Residual-based a posteriori error estimation for hp-adaptive finite element methods for the stokes equations
J. Numer. Math. (IF 3.778) Pub Date : 2019-12-18
Arezou Ghesmati, Wolfgang Bangerth, Bruno Turcksin

Abstract We derive a residual-based a posteriori error estimator for the conforming hp-Adaptive Finite Element Method (hp-AFEM) for the steady state Stokes problem describing the slow motion of an incompressible fluid. This error estimator is obtained by extending the idea of a posteriori error estimation for the classical h-version of AFEM. We also establish the reliability and efficiency of the error

• A Tight Nonlinear Approximation Theory for Time Dependent Closed Quantum Systems
J. Numer. Math. (IF 3.778) Pub Date : 2019-09-25
Joseph W. Jerome

Abstract The approximation of fixed points by numerical fixed points was presented in the elegant monograph of Krasnosel’skii et al. (1972). The theory, both in its formulation and implementation, requires a differential operator calculus, so that its actual application has been selective. The writer and Kerkhoven demonstrated this for the semiconductor drift-diffusion model in 1991. In this article

• POD-ROM for the Darcy-Brinkman Equations with Double-Diffusive Convection
J. Numer. Math. (IF 3.778) Pub Date : 2019-09-25
Fatma G. Eroglu, Songul Kaya, Leo G. Rebholz

Abstract This paper extends proper orthogonal decomposition reduced order modeling to flows governed by double diffusive convection, which models flow driven by two potentials with different rates of diffusion. We propose a reduced model based on proper orthogonal decomposition, present a stability and convergence analyses for it, and give results for numerical tests on a benchmark problem which show

• Discontinuous Galerkin time discretization methods for parabolic problems with linear constraints
J. Numer. Math. (IF 3.778) Pub Date : 2019-09-25
Igor Voulis, Arnold Reusken

Abstract We consider time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. Prototype examples that fit into the general framework are the heat equation with inhomogeneous (time-dependent) Dirichlet boundary conditions and the time-dependent Stokes equation with an inhomogeneous divergence constraint. Two common ways of treating such linear constraints, namely

• Superconvergent discontinuous Galerkin methods for nonlinear parabolic initial and boundary value problems
J. Numer. Math. (IF 3.778) Pub Date : 2019-09-25

Abstract In this article, we discuss error estimates for nonlinear parabolic problems using discontinuous Galerkin methods which include HDG method in the spatial direction while keeping time variable continuous. When piecewise polynomials of degree k ⩾ 1 are used to approximate both the potential as well as the flux, it is shown that the error estimate for the semi-discrete flux in L∞(0, T; L2)-norm

• On Sinc Quadrature Approximations of Fractional Powers of Regularly Accretive Operators
J. Numer. Math. (IF 3.778) Pub Date : 2019-06-26
Andrea Bonito, Wenyu Lei, Joseph E. Pasciak

Abstract We consider the finite element approximation of fractional powers of regularly accretive operators via the Dunford–Taylor integral approach. We use a sinc quadrature scheme to approximate the Balakrishnan representation of the negative powers of the operator as well as its finite element approximation. We improve the exponentially convergent error estimates from [A. Bonito and J. E. Pasciak

• Dual weighted residual error estimation for the finite cell method
J. Numer. Math. (IF 3.778) Pub Date : 2019-06-26
Paolo Di Stolfo, Andreas Rademacher, Andreas Schröder

Abstract The paper presents a goal-oriented error control based on the dual weighted residual method (DWR) for the finite cell method (FCM), which is characterized by an enclosing domain covering the domain of the problem. The error identity derived by the DWR method allows for a combined treatment of the discretization and quadrature error introduced by the FCM. We present an adaptive strategy with

• L2-error analysis of an isoparametric unfitted finite element method for elliptic interface problems
J. Numer. Math. (IF 3.778) Pub Date : 2019-06-26
Christoph Lehrenfeld, Arnold Reusken

Abstract In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method was introduced which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. This method

• Adapted explicit two-step peer methods
J. Numer. Math. (IF 3.778) Pub Date : 2019-06-26
Dajana Conte, Raffaele D’Ambrosio, Martina Moccaldi, Beatrice Paternoster

Abstract In this paper, we present a general class of exponentially fitted two-step peer methods for the numerical integration of ordinary differential equations. The numerical scheme is constructed in order to exploit a-priori known information about the qualitative behaviour of the solution by adapting peer methods already known in literature. Examples of methods with 2 and 3 stages are provided

• Preconditioning methods for eddy-current optimally controlled time-harmonic electromagnetic problems
J. Numer. Math. (IF 3.778) Pub Date : 2019-03-26
Owe Axelsson, Dalibor Lukáš

Abstract Time-harmonic problems arise in many important applications, such as eddy current optimally controlled electromagnetic problems. Eddy current modelling can also be used in non-destructive testings of conducting materials. Using a truncated Fourier series to approximate the solution, for linear problems the equation for different frequencies separate, so it suffices to study solution methods

• Balanced-norm error estimates for sparse grid finite element methods applied to singularly perturbed reaction-diffusion problems
J. Numer. Math. (IF 3.778) Pub Date : 2019-03-26
Stephen Russell, Martin Stynes

Abstract We consider a singularly perturbed linear reaction–diffusion problem posed on the unit square in two dimensions. Standard finite element analyses use an energy norm, but for problems of this type, this norm is too weak to capture adequately the behaviour of the boundary layers that appear in the solution. To address this deficiency, a stronger so-called ‘balanced’ norm has been considered

• Convergence of explicitely coupled Simulation Tools (Co-simulations)
J. Numer. Math. (IF 3.778) Pub Date : 2019-03-26
Thilo Moshagen

Abstract In engineering, it is a common desire to couple existing simulation tools together into one big system by passing information from subsystems as parameters into the subsystems under influence. As executed at fixed time points, this data exchange gives the global method a strong explicit component. Globally, such an explicit co-simulation schemes exchange time step can be seen as a step of

• An efficient preconditioning method for state box-constrained optimal control problems
J. Numer. Math. (IF 3.778) Pub Date : 2018-12-19
Owe Axelsson, Maya Neytcheva, Anders Ström

Abstract An efficient preconditioning technique used earlier for two-by-two block matrix systems with square matrix blocks is shown to be applicable also for a state variable box-constrained optimal control problem. The problem is penalized by a standard regularization term for the control variable and for the box-constraint, using a Moreau–Yosida penalization method. It is shown that there occur very

• The deal.II Library, Version 9.0
J. Numer. Math. (IF 3.778) Pub Date : 2018-12-19
Giovanni Alzetta, Daniel Arndt, Wolfgang Bangerth, Vishal Boddu, Benjamin Brands, Denis Davydov, Rene Gassmöller, Timo Heister, Luca Heltai, Katharina Kormann, Martin Kronbichler, Matthias Maier, Jean-Paul Pelteret, Bruno Turcksin, David Wells

Abstract This paper provides an overview of the new features of the finite element library deal.II version 9.0.

• A Contraction Property of an Adaptive Divergence-Conforming Discontinuous Galerkin Method for the Stokes Problem
J. Numer. Math. (IF 3.778) Pub Date : 2018-12-19
Natasha Sharma, Guido Kanschat

Abstract We prove the contraction property for two successive loops of the adaptive algorithm for the Stokes problem reducing the error of the velocity. The problem is discretized by a divergence-conforming discontinuous Galerkin method which separates pressure and velocity approximation due to its cochain property. This allows us to establish the quasi-orthogonality property which is crucial for the

• Stability and consistency of a finite difference scheme for compressible viscous isentropic flow in multi-dimension
J. Numer. Math. (IF 3.778) Pub Date : 2018-09-25

Abstract Motivated by the work of Karper , we propose a numerical scheme to compressible Navier-Stokes system in spatial multi-dimension based on finite differences. The backward Euler method is applied for the time discretization, while a staggered grid, with continuity and momentum equations on different grids, is used in space. The existence of a solution to the implicit nonlinear scheme, strictly

• A priori error estimates of Adams–Bashforth discontinuous Galerkin methods for scalar nonlinear conservation laws
J. Numer. Math. (IF 3.778) Pub Date : 2018-09-25
Charles Puelz, Béatrice Rivière

Abstract In this paper we show theoretical convergence of a second-order Adams-Bashforth discontinuous Galerkin method for approximating smooth solutions to scalar nonlinear conservation laws with E-fluxes. A priori error estimates are also derived for a first-order forward Euler discontinuous Galerkin method. Rates are optimal in time and suboptimal in space; they are valid under a CFL condition.

• A note on distributionally robust optimization under moment uncertainty
J. Numer. Math. (IF 3.778) Pub Date : 2018-09-25
Qiang Liu, Jia Wu, Xiantao Xiao, Liwei Zhang

Abstract We considers a distributionally robust optimization problem when the ambiguity set specifies the support as well as the mean and the covariance matrix of the uncertain parameters. After deriving a general deterministic reformulation for the distributionally robust optimization problem, we obtain tractable optimization reformulations when the support set is the whole space and when it is a

• Numerical solution of the infinite-dimensional LQR problem and the associated Riccati differential equations
J. Numer. Math. (IF 3.778) Pub Date : 2018-03-26
Peter Benner, Hermann Mena

Abstract The numerical analysis of linear quadratic regulator design problems for parabolic partial differential equations requires solving Riccati equations. In the finite time horizon case, the Riccati differential equation (RDE) arises. The coefficient matrices of the resulting RDE often have a given structure, e.g., sparse, or low-rank. The associated RDE usually is quite stiff, so that implicit

• Mathematical and numerical modeling of plate dynamics with rotational inertia
J. Numer. Math. (IF 3.778) Pub Date : 2018-03-26
Francesco Bonaldi, Giuseppe Geymonat, Françoise Krasucki, Marina Vidrascu

Abstract We give a presentation of the mathematical and numerical treatment of plate dynamics problems including rotational inertia. The presence of rotational inertia in the equation of motion makes the study of such problems interesting. We employ HCT finite elements for space discretization and the Newmark method for time discretization in FreeFEM++, and test such methods in some significant cases:

• Error estimates for higher-order finite volume schemes for convection diffusion problems
J. Numer. Math. (IF 3.778) Pub Date : 2018-03-26
Dietmar Kröner, Mirko Rokyta

Abstract It is still an open problem to prove a priori error estimates for finite volume schemes of higher order MUSCL type, including limiters, on unstructured meshes, which show some improvement compared to first order schemes. In this paper we use these higher order schemes for the discretization of convection dominated elliptic problems in a convex bounded domain Ω in ℝ2 and we can prove such kind

• Mathematical and computational studies of fractional reaction-diffusion system modelling predator-prey interactions
J. Numer. Math. (IF 3.778) Pub Date : 2018-01-29