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Mathematics
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Recent Developments in Numerical Methods for Partial Differential Equations
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Recent Developments in Numerical Methods for Partial Differential Equations

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: 31 May 2026 | Viewed by 1925

Special Issue Editor

Prof. Dr. Olivier Pironneau

E-Mail Website
Guest Editor
Laboratoire Jacques-Louis Lions, Sorbonne Université, 75005 Paris, France
Interests: applied mathematics; numerical methods; partial differential equations; finite element methods; high performance computing; computational fluid dynamics; financial engineering

Special Issue Information

Dear Colleagues,

The time has come to survey the recent developments in the mathematical and practical aspects of analytical and scientific computing for partial differential equations.  As such, we welcome your publications for this Special Issue, “Recent Developments in Numerical Methods for Partial Differential Equations”.

The Issue focuses on the numerical analysis of PDEs, numerical methods for nonlinear or large-scale problems and systems, and computational techniques, including sparse linear solvers, hierarchical methods, and domain decomposition methods. We are also interested in contributions on implementation tools, programming languages, and libraries. etc.  Finally,  artificial intelligence is rapidly becoming an impressive acceleration tool for PDE solvers.

Applications are numerous in physics, engineering, and quantitative finance, amongst other areas. However, applications without algorithmic innovation are not the primary focus of this Special Issue, as other Special Issues already address areas like fluid dynamics and computational physics.

Thank you for your support and interest in contributing to this Special Issue.

Prof. Dr. Olivier Pironneau
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • numerical analysis
  • scientific computing
  • numerical methods
  • high-performance computing
  • partial differential equations

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Published Papers (3 papers)

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Research

26 pages, 2474 KB  
Article
Mathematical Aspects of ANM/FEM Numerical Model, Applied to Nonlinear Elastic, and Thermo Elastic Analysis of Wrinkles in Film/Substrate Systems, and a New Implementation in the FreeFEM++ Language
by Pascal Ventura, Frédéric Hecht, Michel Potier-Ferry, Hamid Zahrouni, Fan Xu, Hamza Azzayani, Michael Brun and Anh-Khoa Chau
Mathematics 2025, 13(19), 3063; https://doi.org/10.3390/math13193063 - 23 Sep 2025
Viewed by 164
Abstract
The main purposes of the present paper are to present the mathematical and algorithmic aspects of the ANM/FEM numerical model and to show how it is applied to analyze elastic and thermo-elastic nonlinear solid mechanical problems. ANM is a robust continuation method based [...] Read more.
The main purposes of the present paper are to present the mathematical and algorithmic aspects of the ANM/FEM numerical model and to show how it is applied to analyze elastic and thermo-elastic nonlinear solid mechanical problems. ANM is a robust continuation method based on a perturbation technique for solving nonlinear problems dependent on a loading parameter. Historically, this technique has been successfully applied to problems in various fields of solid and fluid mechanics. This paper shows how ANM is used to solve nonlinear elastic and nonlinear thermo-elastic problems involving elastic behavior and geometrical nonlinearities. The implementation of ANM for FEM in the FreeFEM++ language is then presented. The FEM software development platform, called FreeFEM++, is structured to work with variational formulations and, therefore, is well adapted to implement ANM for instability problems in solid mechanics. In order to illustrate the great efficiency of FreeFEM++, scripts will be presented for computing the different steps of ANM continuation for solid elastic structures, considering simple geometries subjected to conservative loading. For the purpose of validation, the problem of a cantilever subjected to an applied force is presented. Next, the new numerical model is applied to study wrinkles appearing in a planar film/substrate system that is subjected to compressive surface forces at the lateral faces of the film. Finally, the model is applied to a spherical film/substrate system subjected to thermo-elastic shrinkage. In both cases, the ANM/FEM prediction method, together with a Newton–Riks correction (if needed), identifies the equilibrium paths efficiently, especially after the post-buckling regime. Full article
(This article belongs to the Special Issue Recent Developments in Numerical Methods for Partial Differential Equations)
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17 pages, 464 KB  
Article
A Fokker–Planck Model for Optical Flow Estimation and Image Registration
by Tudor Barbu, Costică Moroşanu and Silviu-Dumitru Pavăl
Mathematics 2025, 13(17), 2807; https://doi.org/10.3390/math13172807 - 1 Sep 2025
Viewed by 342
Abstract
The optical flow problem and image registration problem are treated as optimal control problems associated with Fokker–Planck equations with controller u in the drift term. The payoff is of the form [...] Read more.
The optical flow problem and image registration problem are treated as optimal control problems associated with Fokker–Planck equations with controller u in the drift term. The payoff is of the form 12|y(T)−y1|2+α∫0T|u(t)|44dt, where y1 is the observed final state and y=yu is the solution to the state control system. Here, we prove the existence of a solution and obtain also the Euler–Lagrange optimality conditions which generate a gradient type algorithm for the above optimal control problem. A conceptual algorithm to compute the approximating optimal control and numerical implementation of this algorithm is discussed. Full article
(This article belongs to the Special Issue Recent Developments in Numerical Methods for Partial Differential Equations)
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20 pages, 4194 KB  
Article
Algorithm for Acoustic Wavefield in Space-Wavenumber Domain of Vertically Heterogeneous Media Using NUFFT
by Ying Zhang and Shikun Dai
Mathematics 2025, 13(4), 571; https://doi.org/10.3390/math13040571 - 9 Feb 2025
Viewed by 742
Abstract
Balancing efficiency and accuracy is often challenging in the numerical solution of three-dimensional (3D) point source acoustic wave equations for layered media. To overcome this, an efficient solution method in the spatial-wavenumber domain is proposed, utilizing the Non-Uniform Fast Fourier Transform (NUFFT) to [...] Read more.
Balancing efficiency and accuracy is often challenging in the numerical solution of three-dimensional (3D) point source acoustic wave equations for layered media. To overcome this, an efficient solution method in the spatial-wavenumber domain is proposed, utilizing the Non-Uniform Fast Fourier Transform (NUFFT) to achieve arbitrary non-uniform sampling. By performing a two-dimensional (2D) Fourier transform on the 3D acoustic wave equation in the horizontal direction, the 3D equation is transformed into a one-dimensional (1D) space-wavenumber-domain ordinary differential equation, effectively simplifying significant 3D problems into one-dimensional problems and significantly reducing the demand for memory. The one-dimensional finite-element method is applied to solve the boundary value problem, resulting in a pentadiagonal system of equations. The Thomas algorithm then efficiently solves the system, yielding the layered wavefield distribution in the space-wavenumber domain. Finally, the wavefield distribution in the spatial domain is reconstructed through a 2D inverse Fourier transform. The correctness of the algorithm was verified by comparing it with the finite-element method. The analysis of the half-space model shows that this method can accurately calculate the wavefield distribution in the air layer considering the air layer while exhibiting high efficiency and computational stability in ultra-large-scale models. The three-layer medium model test further verified the adaptability and accuracy of the algorithm in calculating the distribution of acoustic waves in layered media. Through a sensitivity analysis, it is shown that the denser the mesh node partitioning, the higher the medium velocity, and the lower the point source frequency, the higher the accuracy of the algorithm. An algorithm efficiency analysis shows that this method has extremely low memory usage and high computational efficiency and can quickly solve large-scale models even on personal computers. Compared with traditional FEM, the algorithm has much higher advantages in terms of memory usage and efficiency. This method provides a new approach to the numerical solution of partial differential equations. It lays an essential foundation for background field calculation in the scattering seismic numerical simulation and full-waveform inversion of acoustic waves, with strong theoretical significance and practical application value. Full article
(This article belongs to the Special Issue Recent Developments in Numerical Methods for Partial Differential Equations)
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