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Journal of Numerical Mathematics

Editor-in-Chief: Maxim Olshanskii
Managing Editor: Andrea Bonito

Language: English

First published: March 1, 2001

Publication Frequency: 4 issues per year

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The Journal of Numerical Mathematics (formerly East-West Journal of Numerical Mathematics) contains high-quality papers featuring contemporary research in all areas of Numerical Mathematics. This includes the development, analysis, and implementation of new and innovative methods in Numerical Linear Algebra, Numerical Analysis, Optimal Control/Optimization, and Scientific Computing. The journal will also publish applications-oriented papers with significant mathematical content in computational fluid dynamics and other areas of computational engineering, finance, and life sciences.

Topics

Numerical Mathematics
Computational Mathematics
Applied Mathematics
Numerical Linear Algebra
Numerical Analysis
Optimal Control/Optimization
Scientific Computing
Computational Fluid Dynamics
Finance
Life Sciences

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The deal.II library, version 9.7

September 29, 2025

Daniel Arndt, Wolfgang Bangerth, Maximilian Bergbauer, Bruno Blais, Marc Fehling, Rene Gassmöller, Timo Heister, Luca Heltai, Martin Kronbichler, Matthias Maier, Peter Munch, Sam Scheuerman, Bruno Turcksin, Siarhei Uzunbajakau, David Wells, Michał Wichrowski

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This paper provides an overview of the new features of the finite element library deal.II , version 9.7.
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Error analysis for a Crouzeix–Raviart approximation of the obstacle problem

September 22, 2025

Sören Bartels, Alex Kaltenbach

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In the present paper, we study a Crouzeix–Raviart approximation of the obstacle problem, which imposes the obstacle constraint in the midpoints ( i.e. , barycenters) of the elements of a triangulation. We establish a priori error estimates imposing natural regularity assumptions, which are optimal, and the reliability and efficiency of a primal-dual type a posteriori error estimator for general obstacles and involving data oscillation terms stemming only from the right-hand side. Numerical experiments are carried out to support the theoretical findings.
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New analysis of overlapping Schwarz methods for vector field problems in three dimensions with generally shaped domains

September 9, 2025

Duk-Soon Oh, Shangyou Zhang

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This paper introduces a novel approach to analyze two-level overlapping Schwarz methods for Nédélec and Raviart–Thomas vector field problems. The theory is based on new regular stable decompositions for vector fields that are robust to the topology of the domain. Enhanced estimates for the condition numbers of the preconditioned linear systems are derived, dependent linearly on the relative overlap between the overlapping subdomains. Furthermore, we present the numerical experiments which support our theoretical results.
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The Ostrovsky–Hunter equation with a space dependent flux function

August 26, 2025

Neelabja Chatterjee, Nils H. Risebro

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We study the periodic Ostrovsky–Hunter equation in the case where the flux function may depend on the spatial variable. Our main results are that if the flux function is twice differentiable, then there exists a unique entropy solution. This entropy solution may be constructed as a limit of approximate solutions generated by a finite volume scheme, and the finite volume approximations converge to the entropy solution at a rate 1/2.
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An adaptive space-time method for nonlinear poroviscoelastic flows with discontinuous porosities

August 25, 2025

Markus Bachmayr, Simon Boisserée

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This paper is concerned with a space-time adaptive numerical method for instationary porous media flows with nonlinear interaction between porosity and pressure, with focus on problems with discontinuous initial porosities. A convergent method that yields computable error bounds is constructed by a combination of Picard iteration and a least-squares formulation. The adaptive scheme permits spatially variable time steps, which in numerical tests are shown to lead to efficient approximations of solutions with localized porosity waves. The method is also observed to exhibit optimal convergence with respect to the total number of spatio-temporal degrees of freedom.
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C ¹ preserving Hermite interpolants for piecewise polynomials in two dimensions

August 13, 2025

L. Ridgway Scott

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We describe a representation of C 1 piecewise polynomials defined using generalized Hermite elements in two dimensions. This provides a way to generate C 1 piecewise polynomials using nodal variables that are the same at all vertices for degree n ⩾ 5. From this, one obtains both the dimension of the space and an interpolant.
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Exact shape derivatives with unfitted finite element methods

August 7, 2025

Jeremy T. Shahan, Shawn W. Walker

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We present an approach to shape optimization problems that uses an unfitted finite element method (FEM). The domain geometry is represented, and optimized, using a (discrete) level set function and we consider objective functionals that are defined over bulk domains. For a discrete objective functional, defined in the unfitted FEM framework, we show that the exact discrete shape derivative essentially matches the shape derivative at the continuous level. In other words, our approach has the benefits of both optimize-then-discretize and discretize-then-optimize approaches. Specifically, we establish the shape Fréchet differentiability of discrete (unfitted) bulk shape functionals using both the perturbation of the identity approach and direct perturbation of the level set representation. The latter approach is especially convenient for optimizing with respect to level set functions. Moreover, our Fréchet differentiability results hold for any polynomial degree used for the discrete level set representation of the domain. We illustrate our results with some numerical accuracy tests, a simple model (geometric) problem with known exact solution, as well as shape optimization of structural designs.
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Multilevel Picard algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities

August 7, 2025

Ariel Neufeld, Sizhou Wu

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In this paper we introduce a multilevel Picard approximation algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities whose coefficient functions do not need to be constant. We also provide a full convergence and complexity analysis of our algorithm. To obtain our main results, we consider a particular stochastic fixed-point equation (SFPE) motivated by the Feynman–Kac representation and the Bismut–Elworthy–Li formula. We show that the PDE under consideration has a unique viscosity solution which coincides with the first component of the unique solution of the stochastic fixed-point equation. Moreover, the gradient of the unique viscosity solution of the PDE exists and coincides with the second component of the unique solution of the stochastic fixed-point equation. Furthermore, we also provide a numerical example in up to 300 dimensions to demonstrate the practical applicability of our multilevel Picard algorithm.
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Consistency and convergence of flux-corrected finite element methods for nonlinear hyperbolic problems

June 26, 2025

Dmitri Kuzmin, Mária Lukáčová-Medvid’ová, Philipp Öffner

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We investigate the consistency and convergence of flux-corrected finite element approximations in the context of nonlinear hyperbolic conservation laws. In particular, we focus on a monolithic convex limiting approach and prove a Lax–Wendroff-type theorem for the corresponding semi-discrete problem. A key component of our analysis is the use of a weak estimate on bounded variation, which follows from the semi-discrete entropy stability property of the method under investigation. For the Euler equations of gas dynamics, we prove the weak convergence of the flux-corrected finite element scheme to a dissipative weak solution under the assumption that a gas stays in physically reasonable region, i.e., the density is uniformly bounded away from zero, and the energy is uniformly bounded from above. Furthermore, if a strong solution exists, the sequence of numerical approximations converges strongly to the strong solution.
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Improvements of algebraic flux-correction schemes based on Bernstein finite elements

June 26, 2025

Hennes Hajduk

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In Galerkin finite element schemes, the discrete first derivative operator for each spatial dimension is a square matrix that is skew-symmetric under restrictive assumptions for certain types of discretizations and boundary conditions. In most settings, however, this desirable property is violated, often only for a few pairs of nodes. These exceptions can invalidate certain design principles based on the skew-symmetry assumption made for these operators. This paper demonstrates that algebraic manipulations can be performed to make the discrete gradient operators of Bernstein polynomial-based finite element methods skew symmetric. Interest in such discretizations has recently been increasing because they represent natural extensions of second-order algebraic flux correction schemes to higher-order spaces. We employ the new operators in the context of such property-preserving methods, mostly based on discontinuous Galerkin discretizations of arbitrary order. Additional theoretical results for the schemes under investigation are derived, including local and global entropy inequalities, among others. Moreover, a discussion on the optimality of CFL-like time step restrictions arising in explicit Runge–Kutta methods shows that our new approach is superior to earlier representatives of operators employed in similar contexts. These techniques use the monolithic convex limiting paradigm and are applied to the compressible Euler equations.
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SUPG space-time scheme on anisotropic meshes for general parabolic equations

June 4, 2025

Ioannis Toulopoulos

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In this paper, a stabilized space-time finite element scheme on anisotropic quadrilateral meshes for general linear parabolic problems is considered. The scheme is devised on the basis of a unified space-time variational formulation, and uses continuous piece-wise polynomial spaces. The stabilization is achieved by incorporating Streamline-Upwind Petrov–Galerkin (SUPG) techniques. Defining appropriately the stability coefficients, we initially show anisotropic interpolation estimates and then a priori error estimates by following a classical finite element methodology. A series of numerical examples illustrates the theoretical findings.
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Adaptive computation of elliptic eigenvalue topology optimization with a phase-field approach

February 27, 2025

Jing Li, Yifeng Xu, Shengfeng Zhu

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In this paper, we study adaptive approximations of an elliptic eigenvalue optimization problem in a phase-field setting by a conforming finite element method. An adaptive algorithm is proposed and implemented in several two-dimensional numerical examples for illustration of efficiency and accuracy. Theoretical findings consist in the vanishing limit of a subsequence of estimators and the convergence of the relevant subsequence of adaptively-generated solutions to a solution to the continuous optimality system.
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From asymptotic distribution and vague convergence to uniform convergence, with numerical applications

January 1, 2000

Giovanni Barbarino, Sven-Erik Ekström, Carlo Garoni, David Meadon, Stefano Serra-Capizzano, Paris Vassalos

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Let { Λ n = { λ 1 , n , … , λ d n , n } } n ${\left\{{{\Lambda}}_{n}=\left\{{\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right\}\right\}}_{n}$ be a sequence of finite multisets of real numbers such that d n → ∞ as n → ∞, and let f : Ω ⊂ R d → R $f:{\Omega}\subset {\mathbb{R}}^{d}\to \mathbb{R}$ be a Lebesgue measurable function defined on a domain Ω with 0 < μ d (Ω) < ∞, where μ d is the Lebesgue measure in R d ${\mathbb{R}}^{d}$ . We say that { Λ n } n ${\left\{{{\Lambda}}_{n}\right\}}_{n}$ has an asymptotic distribution described by f , and we write { Λ n } n ∼ f ${\left\{{{\Lambda}}_{n}\right\}}_{n}\sim f$ , if (*) lim n → ∞ 1 d n ∑ i = 1 d n F ( λ i , n ) = 1 μ d ( Ω ) ∫ Ω F ( f ( x ) ) d x ${\mathrm{lim}}_{n\to \infty }\frac{1}{{d}_{n}}{\sum }_{i=1}^{{d}_{n}}F\left({\lambda }_{i,n}\right)=\frac{1}{{\mu }_{d}\left({\Omega}\right)}{\int }_{{\Omega}}F\left(f\left(\boldsymbol{x}\right)\right)\mathrm{d}\boldsymbol{x}$ for every continuous function F with bounded support. If Λ n is the spectrum of a matrix A n , we say that { A n } n ${\left\{{A}_{n}\right\}}_{n}$ has an asymptotic spectral distribution described by f and we write { A n } n ∼ λ f ${\left\{{A}_{n}\right\}}_{n}{ \sim }_{\lambda } f$ . In the case where d = 1, Ω is a bounded interval, Λ n ⊆ f (Ω) for all n , and f satisfies suitable conditions, Bogoya, Böttcher, Grudsky, and Maximenko proved that the asymptotic distribution (*) implies the uniform convergence to 0 of the difference between the properly sorted vector [ λ 1 , n , … , λ d n , n ] $\left[{\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right]$ and the vector of samples [ f ( x 1 , n ) , … , f ( x d n , n ) ] $\left[f\left({x}_{1,n}\right),\dots ,f\left({x}_{{d}_{n},n}\right)\right]$ , i.e., (**) lim n → ∞ max i = 1 , … , d n | f ( x i , n ) − λ τ n ( i ) , n | = 0 , ${\mathrm{lim}}_{n\to \infty } {\mathrm{max}}_{i=1,\dots ,{d}_{n}}\vert f\left({x}_{i,n}\right)-{\lambda }_{{\tau }_{n}\left(i\right),n}\vert =0,$ where x 1 , n , … , x d n , n ${x}_{1,n},\dots ,{x}_{{d}_{n},n}$ is a uniform grid in Ω and τ n is the sorting permutation. We extend this result to the case where d ⩾ 1 and Ω is a Peano–Jordan measurable set (i.e., a bounded set with μ d (∂Ω) = 0). We also formulate and prove a uniform convergence result analogous to (**) in the more general case where the function f takes values in the space of k × k matrices. Our derivations are based on the concept of monotone rearrangement (quantile function) as well as on matrix analysis arguments stemming from the theory of generalized locally Toeplitz sequences and the observation that any finite multiset of numbers Λ n = { λ 1 , n , … , λ d n , n } ${{\Lambda}}_{n}=\left\{{\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right\}$ can always be interpreted as the spectrum of a matrix A n = diag ( λ 1 , n , … , λ d n , n ) ${A}_{n}=\mathrm{diag}\left({\lambda }_{1,n},\dots ,{\lambda }_{{d}_{n},n}\right)$ . The theoretical results are illustrated through numerical experiments, and a reinterpretation of them in terms of vague convergence of probability measures is hinted.
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Benchmarking numerical algorithms for harmonic maps into the sphere

January 1, 2000

Sören Bartels, Klaus Böhnlein, Christian Palus, Oliver Sander

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We numerically benchmark methods for computing harmonic maps into the unit sphere, with particular focus on harmonic maps with singularities. For the discretization we compare two different approaches, both based on Lagrange finite elements. While the first method enforces the unit-length constraint only at the Lagrange nodes, the other one adds a pointwise projection to fulfill the constraint everywhere. For the solution of the resulting algebraic problems we compare a nonconforming gradient flow with a Riemannian trust-region method. Both are energy-decreasing and can be shown to converge globally to stationary points of the discretized Dirichlet energy. We observe that while the nonconforming and the conforming discretizations both show similar behavior for smooth problems, the nonconforming discretization handles singularities better. On the solver side, the second-order trust-region method converges after few steps, whereas the number of gradient-flow steps increases proportionally to the inverse grid element diameter.

Journal Impact Factor	4.0	2024, Journal Citation Reports (Clarivate, 2025)
5-year Journal Impact Factor	2.6	2024, Journal Citation Reports (Clarivate, 2025)
Journal Citation Indicator	2.40	2024, Journal Citation Reports (Clarivate, 2025)
CiteScore	6.2	2024, Scopus (Elsevier B.V., 2025)
SCImago Journal Rank	1.638	2024, SJR (Scimago Lab, 2025; Data Source: Scopus)
Source Normalized Impact per Paper	1.898	2024, CWTS Journal Indicators (CWTS B.V., 2025; Data Source: Scopus)
Mathematical Citation Quotient	1.75

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o Articles in journals:　
[1] A. Carbonaro, G. Metafune and C. Spina, Parabolic Schrödinger operators, J. Math
Anal. and Appl. 343 (2008), 965-974.
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[2] D. Farley and L. Sabalka, Presentations of graph braid groups, Forum Math. (2010), doi: 10.1515/FORM.2011.086.
o Books and Monographs:　
[3] I. M. Isaacs, Character theory of finite groups, AMS Chelsea Publishing, 2006.
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[4] T. De Medts, F. Haot, R. Knop and H. Van Maldeghem, On the uniqueness of the unipotent subgroups of some Moufang sets, in: Finite Geometries, Groups, and Computation, pp. 43-66, de Gruyter, Berlin, 2006.

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Editor-in-Chief
Maxim Olshanskii, Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA
molshan@math.uh.edu

Managing Editor
Andrea Bonito, Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
bonito@math.tamu.edu

Editorial Advisory Board

Markus Bachmayr, Institut für Geometrie und Praktische Mathematik
Michele Benzi, Scuola Normale Superiore di Pisa
Susanne Brenner, Louisiana State University
Erik Burman, University College London
Carsten Carstensen, Humboldt University of Berlin
Miloslav Feistauer, Charles University in Prague
Simon Foucart, Texas A&M University
Jean-Luc Guermond, Texas A&M University
Johnny Guzman, Brown University
Daniel Kressner, EPFL
Christoph Lehrenfeld, University of Gottingen
Frédéric Nataf, Sorbonne University
Michael Neilan, University of Pittsburgh
Fabio Nobile, EPFL
Alfio Quarteroni, Politecnico di Milan
Leo Rebholz, Clemson University
Sergey Repin, Russian Academy of Science
Arnold Reusken, RWTH Aachen University
Valeria Simoncini, University of Bologna
Martin Stoll, TU Chemnitz
Eugene Tyrtyshnikov, Russian Academy of Science
Stefan Volkwein, University of Konstanz
Shawn Walker, Louisiana State University
Jun Zou, The Chinese University of Hong Kong

Honorary Members
Pekka Neittaanmäki, University of Jyväskylä
Olivier Pironneau, French Academy of Sciences
Rolf Rannacher, Heidelberg University

Founding Editor
Yuri A. Kuznetsov †

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eISSN:

1569-3953

ISSN:

1570-2820

Language:

English

Publisher:

De Gruyter

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First published:

March 1, 2001

Publication Frequency:

4 issues per year

eISSN: 1569-3953

ISSN: 1570-2820

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