样式: 排序: IF: - GO 导出 标记为已读
-
Exploring numerical blow-up phenomena for the Keller–Segel–Navier–Stokes equations J. Numer. Math. (IF 3.0) Pub Date : 2023-10-16 Jesús Bonilla, Juan Vicente Gutiérrez-Santacreu
The Keller-Segel-Navier-Stokes system governs chemotaxis in liquid environments. This system is to be solved for the organism and chemoattractant densities and for the fluid velocity and pressure. It is known that if the total initial organism density mass is below 2π there exist globally defined generalised solutions, but what is less understood is whether there are blow-up solutions beyond such a
-
Efficient numerical solution of the Fokker-Planck equation using physics-conforming finite element methods J. Numer. Math. (IF 3.0) Pub Date : 2023-09-25 Katharina Wegener, Dmitri Kuzmin, Stefan Turek
We consider the Fokker–Planck equation (FPE) for the orientation probability density of fiber suspensions. Using the continuous Galerkin method, we express the numerical solution in terms of Lagrange basis functions that are associated with N nodes of a computational mesh for a domain in the 3D physical space and M nodes of a mesh for the surface of a unit sphere representing the configuration space
-
Local parameter selection in the C0 interior penalty method for the biharmonic equation J. Numer. Math. (IF 3.0) Pub Date : 2023-09-11 Philipp Bringmann, Carsten Carstensen, Julian Streitberger
The symmetric 0 interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant
-
On the discrete Sobolev inequalities J. Numer. Math. (IF 3.0) Pub Date : 2023-09-05 Sedrick Kameni Ngwamou, Michael Ndjinga
We prove a discrete version of the famous Sobolev inequalities [1] in R d for d ∈ N ∗ , p ∈ [ 1 , + ∞ [ $\mathbb{R}^{d} \text { for } d \in \mathbb{N}^{*}, p \in[1,+\infty[$ for general non orthogonal meshes with possibly non convex cells. We follow closely the proof of the continuous Sobolev inequality based on the embedding of B V R d into L d d − 1 $B V\left(\mathbb{R}^{d}\right) \text { into }
-
Stability and convergence of relaxed scalar auxiliary variable schemes for Cahn–Hilliard systems with bounded mass source J. Numer. Math. (IF 3.0) Pub Date : 2023-08-31 Kei Fong Lam, Ru Wang
The scalar auxiliary variable (SAV) approach of Shen et al. (2018), which presents a novel way to discretize a large class of gradient flows, has been extended and improved by many authors for general dissipative systems. In this work we consider a Cahn–Hilliard system with mass source that, for image processing and biological applications, may not admit a dissipative structure involving the Ginzburg–Landau
-
POD-ROMs for incompressible flows including snapshots of the temporal derivative of the full order solution: Error bounds for the pressure J. Numer. Math. (IF 3.0) Pub Date : 2023-08-26 Bosco García-Archilla, Volker John, Sarah Katz, Julia Novo
Reduced order methods (ROMs) for the incompressible Navier–Stokes equations, based on proper orthogonal decomposition (POD), are studied that include snapshots which approach the temporal derivative of the velocity from a full order mixed finite element method (FOM). In addition, the set of snapshots contains the mean velocity of the FOM. Both the FOM and the POD-ROM are equipped with a grad-div stabilization
-
Fundamental Theory and R-linear Convergence of Stretch Energy Minimization for Spherical Equiareal Parameterization J. Numer. Math. (IF 3.0) Pub Date : 2023-08-24 Tsung-Ming Huang, Wei-Hung Liao, Wen-Wei Lin
Here, we extend the finite distortion problem from bounded domains in ℝ2 to closed genus-zero surfaces in ℝ3 by a stereographic projection. Then, we derive a theoretical foundation for spherical equiareal parameterization between a simply connected closed surface M and a unit sphere 𝕊2 by minimizing the total area distortion energy on ̅ℂ. After the minimizer of the total area distortion energy is
-
The deal.II Library, Version 9.5 J. Numer. Math. (IF 3.0) Pub Date : 2023-08-22 Daniel Arndt, Wolfgang Bangerth, Maximilian Bergbauer, Marco Feder, Marc Fehling, Johannes Heinz, Timo Heister, Luca Heltai, Martin Kronbichler, Matthias Maier, Peter Munch, Jean-Paul Pelteret, Bruno Turcksin, David Wells, Stefano Zampini
This paper provides an overview of the new features of the finite element library deal.II, version 9.5.
-
A posteriori error estimate for a WG method of H(curl)-elliptic problems J. Numer. Math. (IF 3.0) Pub Date : 2023-08-22 Jie Peng, Yingying Xie, Liuqiang Zhong
This paper presents a posteriori error estimate for the weak Galerkin (WG) finite element method used to solve H(curl)-elliptic problems. Firstly, we introduce a WG method for solving H(curl)-elliptic problems and a corresponding residual type error estimator without a stabilization term. Secondly, we establish the reliability of the error estimator by demonstrating that the stabilization term is controlled
-
Error analysis for a Crouzeix–Raviart approximation of the p-Dirichlet problem J. Numer. Math. (IF 3.0) Pub Date : 2023-08-21 Alex Kaltenbach
In the present paper, we examine a Crouzeix–Raviart approximation for non-linear partial differential equations having a (p, δ)-structure for some p ∈ (1, ∞) and δ⩾0. We establish a priori error estimates, which are optimal for all p ∈ (1, ∞) and δ⩾0, medius error estimates, i.e., best-approximation results, and a primal-dual a posteriori error estimate, which is both reliable and efficient. The theoretical
-
Entropy stable non-oscillatory fluxes: An optimized wedding of entropy conservative flux with non-oscillatory flux J. Numer. Math. (IF 3.0) Pub Date : 2023-08-16 Ritesh Kumar Dubey
This work frames the problem of constructing non-oscillatory entropy stable fluxes as a least square optimization problem. A flux sign stability condition is defined for a pair of entropy conservative flux (F∗ ) and a non-oscillatory flux (Fs ). This novel approach paves a way to construct non-oscillatory entropy stable flux (F̂) as a simple combination of (F∗ and Fs ) which inherently optimize the
-
High order immersed hybridized difference methods for elliptic interface problems J. Numer. Math. (IF 3.0) Pub Date : 2023-08-16 Youngmok Jeon
We propose high order conforming and nonconforming immersed hybridized difference (IHD) methods in two and three dimensions for elliptic interface problems. Introducing the virtual to real transformation (VRT), we could obtain a systematic and unique way of deriving arbitrary high order methods in principle. The optimal number of collocating points for imposing interface conditions is proved, and a
-
Diffusion of tangential tensor fields: numerical issues and influence of geometric properties J. Numer. Math. (IF 3.0) Pub Date : 2023-08-15 E. Bachini, P. Brandner, T. Jankuhn, M. Nestler, S. Praetorius, A. Reusken, A. Voigt
We study the diffusion of tangential tensor-valued data on curved surfaces. For this purpose, several finite-element-based numerical methods are collected and used to solve a tangential surface n-tensor heat flow problem. These methods differ with respect to the surface representation used, the geometric information required, and the treatment of the tangentiality condition. We emphasize the importance
-
New non-augmented mixed finite element methods for the Navier–Stokes–Brinkman equations using Banach spaces J. Numer. Math. (IF 3.0) Pub Date : 2023-03-24 Gabriel N. Gatica, Nicolás Núñez, Ricardo Ruiz-Baier
In this paper we consider the Navier–Stokes–Brinkman equations, which constitute one of the most common nonlinear models utilized to simulate viscous fluids through porous media, and propose and analyze a Banach spaces-based approach yielding new mixed finite element methods for its numerical solution. In addition to the velocity and pressure, the strain rate tensor, the vorticity, and the stress tensor
-
Unifying a posteriori error analysis of five piecewise quadratic discretisations for the biharmonic equation J. Numer. Math. (IF 3.0) Pub Date : 2023-01-31 Carsten Carstensen, Benedikt Gräßle, Neela Nataraj
An abstract property (H) is the key to a complete a priori error analysis in the (discrete) energy norm for several nonstandard finite element methods in the recent work [Lowest-order equivalent nonstandard finite element methods for biharmonic plates, Carstensen and Nataraj, M2AN, 2022]. This paper investigates the impact of (H) to the a posteriori error analysis and establishes known and novel explicit
-
Adaptive POD-DEIM correction for Turing pattern approximation in reaction-diffusion PDE systems J. Numer. Math. (IF 3.0) Pub Date : 2023-01-19 Alessandro Alla, Angela Monti, Ivonne Sgura
We investigate a suitable application of Model Order Reduction (MOR) techniques for the numerical approximation of Turing patterns, that are stationary solutions of reaction-diffusion PDE (RD-PDE) systems. We show that solutions of surrogate models built by classical Proper Orthogonal Decomposition (POD) exhibit an unstable error behaviour over the dimension of the reduced space. To overcome this drawback
-
Transformed primal-dual methods for nonlinear saddle point systems J. Numer. Math. (IF 3.0) Pub Date : 2023-01-14 Long Chen, Jingrong Wei
A transformed primal-dual (TPD) flow is developed for a class of nonlinear smooth saddle point system. The flow for the dual variable contains a Schur complement which is strongly convex. Exponential stability of the saddle point is obtained by showing the strong Lyapunov property. Several TPD iterations are derived by implicit Euler, explicit Euler, implicit-explicit and Gauss-Seidel methods with
-
Diagonally implicit Runge-Kutta schemes: Discrete energy-balance laws and compactness properties J. Numer. Math. (IF 3.0) Pub Date : 2022-12-24 Abner J. Salgado, Ignacio Tomas
We study diagonally implicit Runge-Kutta (DIRK) schemes when applied to abstract evolution problems that fit into the Gelfand-triple framework. We introduce novel stability notions that are well-suited to this setting and provide simple, necessary and sufficient, conditions to verify that a DIRK scheme is stable in our sense and in Bochner-type norms. We use several popular DIRK schemes in order to
-
A posteriori error estimates for hierarchical mixed-dimensional elliptic equations J. Numer. Math. (IF 3.0) Pub Date : 2022-10-20 Jhabriel Varela, Elyes Ahmed, Eirik Keilegavlen, Jan M. Nordbotten, Florin A. Radu
Mixed-dimensional elliptic equations exhibiting a hierarchical structure are commonly used to model problems with high aspect ratio inclusions, such as flow in fractured porous media. We derive general abstract estimates based on the theory of functional a posteriori error estimates, for which guaranteed upper bounds for the primal and dual variables and two-sided bounds for the primal-dual pair are
-
A subspace of linear nonconforming finite element for nearly incompressible elasticity and stokes flow J. Numer. Math. (IF 3.0) Pub Date : 2022-09-13 Shangyou Zhang
The linear nonconforming finite element, combined with constant finite element for pressure, is stable for the Stokes problem. But it does not satisfy the discrete Korn inequality. The linear conforming finite element satisfies the discrete Korn inequality, but is not stable for the Stokes problem and fails for the nearly incompressible elasticity problems. We enrich the linear conforming finite element
-
An all Mach number finite volume method for isentropic two-phase flow J. Numer. Math. (IF 3.0) Pub Date : 2022-08-02 Mária Lukáčová-Medvid’ová, Gabriella Puppo, Andrea Thomann
We present an implicit-explicit finite volume scheme for isentropic two phase flow in all Mach number regimes. The underlying model belongs to the class of symmetric hyperbolic thermodynamically compatible models. The key element of the scheme consists of a linearisation of pressure and enthalpy terms at a reference state. The resulting stiff linear parts are integrated implicitly, whereas the non-linear
-
The deal.II Library, Version 9.4 J. Numer. Math. (IF 3.0) Pub Date : 2022-07-16 Daniel Arndt, Wolfgang Bangerth, Marco Feder, Marc Fehling, Rene Gassmöller, Timo Heister, Luca Heltai, Martin Kronbichler, Matthias Maier, Peter Munch, Jean-Paul Pelteret, Simon Sticko, Bruno Turcksin, David Wells
This paper provides an overview of the new features of the finite element library deal.II, version 9.4.
-
A structure preserving front tracking finite element method for the Mullins–Sekerka problem J. Numer. Math. (IF 3.0) Pub Date : 2022-07-15 Robert Nürnberg
We introduce and analyse a fully discrete approximation for a mathematical model for the solidification and liquidation of materials of negligible specific heat. The model is a two-sided Mullins–Sekerka problem. The discretization uses finite elements in space and an independent parameterization of the moving free boundary. We prove unconditional stability and exact volume conservation for the introduced
-
Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations J. Numer. Math. (IF 3.0) Pub Date : 2022-07-12 Martin Hutzenthaler, Arnulf Jentzen, Thomas Kruse, Tuan Anh Nguyen
Backward stochastic differential equations (BSDEs) belong nowadays to the most frequently studied equations in stochastic analysis and computational stochastics. BSDEs in applications are often nonlinear and high-dimensional. In nearly all cases such nonlinear high-dimensional BSDEs cannot be solved explicitly and it has been and still is a very active topic of research to design and analyze numerical
-
Regularity results and numerical solution by the discontinuous Galerkin method to semilinear parabolic initial boundary value problems with nonlinear Newton boundary conditions in a polygonal space-time cylinder J. Numer. Math. (IF 3.0) Pub Date : 2022-06-24 Monika Balázsová, Miloslav Feistauer, Anna-Margarete Sändig
In this note we consider a parabolic evolution equation in a polygonal space-time cylinder. We show, that the elliptic part is given by a m-accretive mapping from L q (Ω) → L q (Ω). Therefore we can apply the theory of nonlinear semigroups in Banach spaces in order to get regularity results in time and space. The second part of the paper deals with the numerical solution of the problem. It is dedicated
-
Fourier analysis of a time-simultaneous two-grid algorithm using a damped Jacobi waveform relaxation smoother for the one-dimensional heat equation J. Numer. Math. (IF 3.0) Pub Date : 2022-06-04 Christoph Lohmann, Jonas Dünnebacke, Stefan Turek
In this work, the convergence behavior of a time-simultaneous two-grid algorithm for the one-dimensional heat equation is studied using Fourier arguments in space. The underlying linear system of equations is obtained by a finite element or finite difference approximation in space while the semi-discrete problem is discretized in time using the θ-scheme. The simultaneous treatment of all time instances
-
A time-explicit weak Galerkin scheme for parabolic equations on polytopal partitions J. Numer. Math. (IF 3.0) Pub Date : 2022-05-31 Junping Wang, Xiu Ye, Shangyou Zhang
In this paper a time-explicit weak Galerkin finite element method is introduced and analyzed for parabolic equations. The main idea relies on the inclusion of a stabilization term in the temporal direction in addition to the usual static stabilization in the weak Galerkin framework. Both semi-discrete and fully-discrete schemes in time are presented, as well as their stability and error analysis. Numerical
-
How to prove optimal convergence rates for adaptive least-squares finite element methods∗ J. Numer. Math. (IF 3.0) Pub Date : 2022-05-04 Philipp Bringmann
The convergence analysis with rates for adaptive least-squares finite element methods (ALSFEMs) combines arguments from the a posteriori analysis of conforming and mixed finite element schemes. This paper provides an overview of the key arguments for the verification of the axioms of adaptivity for an ALSFEM for the solution of a linear model problem. The formulation at hand allows for the simultaneous
-
Numerical analysis for a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport J. Numer. Math. (IF 3.0) Pub Date : 2022-05-04 Harald Garcke, Dennis Trautwein
In this work, we consider a diffuse interface model for tumour growth in the presence of a nutrient which is consumed by the tumour. The system of equations consists of a Cahn–Hilliard equation with source terms for the tumour cells and a reaction-diffusion equation for the nutrient. We introduce a fully-discrete finite element approximation of the model and prove stability bounds for the discrete
-
A posteriori error analysis of Banach spaces-based fully-mixed finite element methods for Boussinesq-type models∗ J. Numer. Math. (IF 3.0) Pub Date : 2022-04-14 Gabriel N. Gatica, Cristian Inzunza, Ricardo Ruiz-Baier, Felipe Sandoval
In this paper we consider Banach spaces-based fully-mixed variational formulations recently proposed for the Boussinesq and the Oberbeck-Boussinesq models, and develop reliable and efficient residual-based a posteriori error estimators for the 2D and 3D versions of the associated mixed finite element schemes. For the reliability analysis, we employ the global inf-sup condition for each sub-model, namely
-
A Divergence-free finite element method for the stokes problem with boundary correction J. Numer. Math. (IF 3.0) Pub Date : 2022-04-14 Haoran Liu, Michael Neilan, M. Baris Otus
This paper constructs and analyzes a boundary correction finite element method for the Stokes problem based on the Scott-Vogelius pair on Clough-Tocher splits. The velocity space consists of continuous piecewise polynomials of degree k, and the pressure space consists of piecewise polynomials of degree (k − 1) without continuity constraints. A Lagrange multiplier space that consists of continuous piecewise
-
An Assessment of Solvers for Algebraically Stabilized Discretizations of Convection-Diffusion-Reaction Equations J. Numer. Math. (IF 3.0) Pub Date : 2022-04-14 Abhinav Jha, Ondřej Pártl, Naveed Ahmed, Dmitri Kuzmin
We consider flux-corrected finite element discretizations of 3D convection-dominated transport problems and assess the computational efficiency of algorithms based on such approximations. The methods under investigation include flux-corrected transport schemes and monolithic limiters. We discretize in space using a continuous Galerkin method and P1 or Q1 finite elements. Time integration is performed
-
Frontmatter J. Numer. Math. (IF 3.0) Pub Date : 2022-03-01
Article Frontmatter was published on March 1, 2022 in the journal Journal of Numerical Mathematics (volume 30, issue 1).
-
POD-Galerkin model order reduction for parametrized nonlinear time-dependent optimal flow control: an application to shallow water equations J. Numer. Math. (IF 3.0) Pub Date : 2022-03-01 Maria Strazzullo, Francesco Ballarin, Gianluigi Rozza
In the present paper we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable, e.g., in a marine environmental monitoring plan
-
Numerical simulation for European and American option of risks in climate change of Three Gorges Reservoir Area J. Numer. Math. (IF 3.0) Pub Date : 2022-03-01 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
-
Two mixed finite element formulations for the weak imposition of the Neumann boundary conditions for the Darcy flow J. Numer. Math. (IF 3.0) Pub Date : 2022-01-18 Erik Burman, Riccardo Puppi
We propose two different discrete formulations for the weak imposition of the Neumann boundary conditions of the Darcy flow. The Raviart-Thomas mixed finite element on both triangular, quadrilateral meshes is considered for both methods. One is a consistent discretization depending on a weighting parameter scaling as O ( h − 1 ) $ \mathcal O(h^{-1}) $ , while the other is a penalty-type formulation
-
A C 0 conforming dg finite element method for biharmonic equations on triangle/tetrahedron J. Numer. Math. (IF 3.0) Pub Date : 2022-01-01 Xiu Ye, Shangyou Zhang
A C 0 conforming discontinuous Galerkin (CDG) finite element method is introduced for solving the biharmonic equation. The first strong gradient of C 0 finite element functions is a vector of discontinuous piecewise polynomials. The second gradient is the weak gradient of discontinuous piecewise polynomials. This method, by its name, uses nonconforming (non C 1) approximations and keeps simple formulation
-
Frontmatter J. Numer. Math. (IF 3.0) Pub Date : 2021-12-01
Article Frontmatter was published on December 1, 2021 in the journal Journal of Numerical Mathematics (volume 29, issue 4).
-
A reduced basis method for fractional diffusion operators II J. Numer. Math. (IF 3.0) Pub Date : 2021-12-01 Tobias Danczul, Joachim Schöberl
We present a novel numerical scheme to approximate the solution map s ↦ u ( s ) := 𝓛 – s f to fractional PDEs involving elliptic operators. Reinterpreting 𝓛 – s as an interpolation operator allows us to write u ( s ) as an integral including solutions to a parametrized family of local PDEs. We propose a reduced basis strategy on top of a finite element method to approximate its integrand. Unlike
-
Schur complement spectral bounds for large hybrid FETI-DP clusters and huge three-dimensional scalar problems J. Numer. Math. (IF 3.0) Pub Date : 2021-12-01 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
-
Acceleration of nonlinear solvers for natural convection problems J. Numer. Math. (IF 3.0) Pub Date : 2021-12-01 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
-
Entropy stabilization and property-preserving limiters for ℙ1 discontinuous Galerkin discretizations of scalar hyperbolic problems J. Numer. Math. (IF 3.0) Pub Date : 2021-12-01 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
-
On Rational Krylov and Reduced Basis Methods for Fractional Diffusion J. Numer. Math. (IF 3.0) Pub Date : 2021-11-14 Tobias Danczul, Clemens Hofreither
We establish an equivalence between two classes of methods for solving fractional diffusion problems, namely, Reduced Basis Methods (RBM) and Rational Krylov Methods (RKM). In particular, we demonstrate that several recently proposed RBMs for fractional diffusion can be interpreted as RKMs. This changed point of view allows us to give convergence proofs for some methods where none were previously available
-
Guaranteed Upper Bounds For The Velocity Error Of Pressure-Robust Stokes Discretisations J. Numer. Math. (IF 3.0) Pub Date : 2021-11-09 P.L. Lederer, C. Merdon
This paper aims to improve guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e. for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager-Synge type result relates the velocity errors of divergence-free primal and perfectly equilibrated dual mixed methods for the velocity stress. The first main result of the paper is
-
An energy, momentum, and angular momentum conserving scheme for a regularization model of incompressible flow J. Numer. Math. (IF 3.0) Pub Date : 2021-11-09 Sean Ingimarson
We introduce a new regularization model for incompressible fluid flow, which is a regularization of the EMAC (energy, momentum, and angular momentum conserving) formulation of the Navier–Stokes equations (NSE) that we call EMAC-Reg. The EMAC formulation has proved to be a useful formulation because it conserves energy, momentum, and angular momentum even when the divergence constraint is only weakly
-
Adaptive space-time finite element methods for parabolic optimal control problems J. Numer. Math. (IF 3.0) Pub Date : 2021-11-09 Ulrich Langer, Andreas Schafelner
We present, analyze, and test locally stabilized space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of space-time tracking parabolic optimal control problems with the standard L 2-regularization. We derive a priori discretization error estimates in terms of the local mesh-sizes for shape-regular meshes. The adaptive version is driven by local
-
A posteriori error analysis of an enriched Galerkin method of order one for the Stokes problem J. Numer. Math. (IF 3.0) Pub Date : 2021-11-09 Vivette Girault, María González, Frédéric Hecht
We derive optimal reliability and efficiency of a posteriori error estimates for the steady Stokes problem, with a nonhomogeneous Dirichlet boundary condition, solved by a stable enriched Galerkin scheme (EG) of order one on triangular or quadrilateral meshes in ℝ2, and tetrahedral or hexahedral meshes in ℝ3.
-
Error analysis for a vorticity/Bernoulli pressure formulation for the Oseen equations J. Numer. Math. (IF 3.0) Pub Date : 2021-08-27 Verónica Anaya, David Mora, Amiya K. Pani, Ricardo Ruiz-Baier
A variational formulation is analysed for the Oseen equations written in terms of vorticity and Bernoulli pressure. The velocity is fully decoupled using the momentum balance equation, and it is later recovered by a post-process. A finite element method is also proposed, consisting in equal-order Nédélec finite elements and piecewise continuous polynomials for the vorticity and the Bernoulli pressure
-
The deal.II library, Version 9.3 J. Numer. Math. (IF 3.0) Pub Date : 2021-08-01 Daniel Arndt, Wolfgang Bangerth, Bruno Blais, Marc Fehling, Rene Gassmöller, Timo Heister, Luca Heltai, Uwe Köcher, Martin Kronbichler, Matthias Maier, Peter Munch, Jean-Paul Pelteret, Sebastian Proell, Konrad Simon, Bruno Turcksin, David Wells, Jiaqi Zhang
This paper provides an overview of the new features of the finite element library deal.II, version 9.3.
-
Analytic Integration of the Newton Potential over Cuboids and an Application to Fast Multipole Methods J. Numer. Math. (IF 3.0) Pub Date : 2021-08-01 Matthias Kirchhart, Donat Weniger
We present simplified formulæ for the analytic integration of the Newton potential of polynomials over boxes in two- and three-dimensional space. These are implemented in an easy-to-use C++ library that allows computations in arbitrary precision arithmetic which is also documented here. We describe how these results can be combined with fast multipole methods to evaluate the Newton potential of more
-
A finite element method for degenerate two-phase flow in porous media. Part I: Well-posedness J. Numer. Math. (IF 3.0) 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 3.0) 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 3.0) 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 3.0) 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 3.0) 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
-
Mixed-hybrid and mixed-discontinuous Galerkin methods for linear dynamical elastic–viscoelastic composite structures J. Numer. Math. (IF 3.0) Pub Date : 2021-07-04 Antonio Márquez, Salim Meddahi
We introduce and analyze a stress-based formulation for Zener’s model in linear viscoelasticity. The method is aimed to tackle efficiently heterogeneous materials that admit purely elastic and viscoelastic parts in their composition.We write the mixed variational formulation of the problem in terms of a class of tensorial wave equation and obtain an energy estimate that guaranties the well-posedness
-
Frontmatter J. Numer. Math. (IF 3.0) 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 3.0) Pub Date : 2021-03-01 Roberto Díaz-Adame, Silvia Jerez
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 3.0) 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 [15], 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 3.0) 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