Pub Date : 2026-05-01Epub Date: 2025-11-18DOI: 10.1016/j.matcom.2025.11.020
Ziyad AlSharawi , Jose S. Cánovas
In this paper, we focus on finding one-dimensional maps that detect global stability in multidimensional maps. We consider various local and global stability techniques in discrete-time dynamical systems and discuss their advantages and limitations. Specifically, we navigate through the embedding technique, the expansion strategy, the dominance condition technique, and the enveloping technique to establish a unifying approach to global stability. We introduce the concept of strong local asymptotic stability (SLAS), then integrate what we call the expansion strategy with the enveloping technique to develop the enveloping technique for two-dimensional maps, which allows to give novel global stability results. Our results make it possible to verify global stability geometrically for two-dimensional maps. We provide several illustrative examples to elucidate our concepts, bolster our theory, and demonstrate its application.
{"title":"Integrating the enveloping technique with the expansion strategy to establish stability","authors":"Ziyad AlSharawi , Jose S. Cánovas","doi":"10.1016/j.matcom.2025.11.020","DOIUrl":"10.1016/j.matcom.2025.11.020","url":null,"abstract":"<div><div>In this paper, we focus on finding one-dimensional maps that detect global stability in multidimensional maps. We consider various local and global stability techniques in discrete-time dynamical systems and discuss their advantages and limitations. Specifically, we navigate through the embedding technique, the expansion strategy, the dominance condition technique, and the enveloping technique to establish a unifying approach to global stability. We introduce the concept of strong local asymptotic stability (SLAS), then integrate what we call the expansion strategy with the enveloping technique to develop the enveloping technique for two-dimensional maps, which allows to give novel global stability results. Our results make it possible to verify global stability geometrically for two-dimensional maps. We provide several illustrative examples to elucidate our concepts, bolster our theory, and demonstrate its application.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"243 ","pages":"Pages 1-15"},"PeriodicalIF":4.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145555218","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-05-01Epub Date: 2025-11-26DOI: 10.1016/j.matcom.2025.11.033
Pedro R.S. Antunes , Hernani Calunga , Pedro Serranho
The MFS-SVD approach introduced in [1], which combines the method of fundamental solutions (MFS) with singular value decomposition (SVD), is a potential alternative to existing methods for improving the conditioning of MFS linear systems. However, until now, the feasibility of this approach for boundary value problems (BVPs) defined on planar domains, has only been illustrated for two problems involving second order partial differential equations: The Laplace equation in [1] and the homogeneous Helmholtz equation in [2]. These papers suggest that SVD should be applied to the ill-conditioned factor of the MFS linear systems decomposition, which does not apply to higher order problems. In this work, we bring more clarity to this point, contributing to the establishment of a complete procedure to follow when solving problems using the MFS-SVD approach. We use the biharmonic boundary value problem, a fourth order PDE, to illustrate this procedure. This is done by approaching the numerical solution of the problem using two different ansatz, which means two different addition theorems and two different decompositions, with the aim of reinforcing the idea about the robustness of the MFS-SVD, regardless of the numerical formulation being considered. As expected, the MFS-SVD performs similarly in both cases.
{"title":"On improving the conditioning of the method of fundamental solutions for biharmonic BVPs in 2D domains","authors":"Pedro R.S. Antunes , Hernani Calunga , Pedro Serranho","doi":"10.1016/j.matcom.2025.11.033","DOIUrl":"10.1016/j.matcom.2025.11.033","url":null,"abstract":"<div><div>The MFS-SVD approach introduced in <span><span>[1]</span></span>, which combines the method of fundamental solutions (MFS) with singular value decomposition (SVD), is a potential alternative to existing methods for improving the conditioning of MFS linear systems. However, until now, the feasibility of this approach for boundary value problems (BVPs) defined on planar domains, has only been illustrated for two problems involving second order partial differential equations: The Laplace equation in <span><span>[1]</span></span> and the homogeneous Helmholtz equation in <span><span>[2]</span></span>. These papers suggest that SVD should be applied to the ill-conditioned factor of the MFS linear systems decomposition, which does not apply to higher order problems. In this work, we bring more clarity to this point, contributing to the establishment of a complete procedure to follow when solving problems using the MFS-SVD approach. We use the biharmonic boundary value problem, a fourth order PDE, to illustrate this procedure. This is done by approaching the numerical solution of the problem using two different ansatz, which means two different addition theorems and two different decompositions, with the aim of reinforcing the idea about the robustness of the MFS-SVD, regardless of the numerical formulation being considered. As expected, the MFS-SVD performs similarly in both cases.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"243 ","pages":"Pages 237-250"},"PeriodicalIF":4.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-05-01Epub Date: 2025-11-25DOI: 10.1016/j.matcom.2025.11.028
Md. Mutakabbir Khan
This work investigates the nonlinear dynamics of a discrete predator–prey system with prey cannibalism and group defense. The model combines Smith-type growth with a cannibalistic term for prey, while predators follow a Monod–Haldane response. Using the center manifold theorem, we establish conditions for period-doubling (PD) and Neimark–Sacker (NS) bifurcations within the biologically feasible region. Numerical simulations validate these theoretical results and reveal complex dynamics, including high-periodic orbits, quasi-periodic invariant closed curves, and chaotic attractors confirmed through maximal Lyapunov exponents. To suppress chaotic fluctuations and restore ecological balance, we implement both the Ott–Grebogi–Yorke (OGY) method and a state feedback control strategy, successfully stabilizing the system near unstable equilibria. This work deepens the understanding of nonlinear mechanisms governing ecological interactions and offers robust control strategies to manage chaos in discrete biological systems.
{"title":"Nonlinear dynamics and Chaos control in a discrete predator–prey model with Smith-type growth, cannibalism, and group defense","authors":"Md. Mutakabbir Khan","doi":"10.1016/j.matcom.2025.11.028","DOIUrl":"10.1016/j.matcom.2025.11.028","url":null,"abstract":"<div><div>This work investigates the nonlinear dynamics of a discrete predator–prey system with prey cannibalism and group defense. The model combines Smith-type growth with a cannibalistic term for prey, while predators follow a Monod–Haldane response. Using the center manifold theorem, we establish conditions for period-doubling (PD) and Neimark–Sacker (NS) bifurcations within the biologically feasible region. Numerical simulations validate these theoretical results and reveal complex dynamics, including high-periodic orbits, quasi-periodic invariant closed curves, and chaotic attractors confirmed through maximal Lyapunov exponents. To suppress chaotic fluctuations and restore ecological balance, we implement both the Ott–Grebogi–Yorke (OGY) method and a state feedback control strategy, successfully stabilizing the system near unstable equilibria. This work deepens the understanding of nonlinear mechanisms governing ecological interactions and offers robust control strategies to manage chaos in discrete biological systems.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"243 ","pages":"Pages 149-170"},"PeriodicalIF":4.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-05-01Epub Date: 2025-12-02DOI: 10.1016/j.matcom.2025.11.036
M. Latha Maheswari , R. Nandhini , Mohammad Sajid
This study investigates the existence and uniqueness of the solution for the boundary value problem (BVP) involving integro-impulsive delay differential equations with Caputo fractional derivatives. The problem incorporates nonlocal and Riemann–Liouville integral boundary conditions. Using the Banach contraction principle, we demonstrate the existence of a unique solution for this fractional BVP. Additionally, we provided numerical examples with graphical representations to illustrate and validate the theoretical results.
{"title":"Exploration on a class of impulsive delay integro-differential systems with fractional boundary conditions","authors":"M. Latha Maheswari , R. Nandhini , Mohammad Sajid","doi":"10.1016/j.matcom.2025.11.036","DOIUrl":"10.1016/j.matcom.2025.11.036","url":null,"abstract":"<div><div>This study investigates the existence and uniqueness of the solution for the boundary value problem (BVP) involving integro-impulsive delay differential equations with Caputo fractional derivatives. The problem incorporates nonlocal and Riemann–Liouville integral boundary conditions. Using the Banach contraction principle, we demonstrate the existence of a unique solution for this fractional BVP. Additionally, we provided numerical examples with graphical representations to illustrate and validate the theoretical results.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"243 ","pages":"Pages 327-338"},"PeriodicalIF":4.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-05-01Epub Date: 2025-11-19DOI: 10.1016/j.matcom.2025.09.025
Yovan Singh , Bapan Ghosh , Suman Mondal
Time delays are integral to ecological processes. Population models incorporate time delays to account for the time required for maturation, gestation, dispersal, and many more. Time delay can induce various stability dynamics, including (i) stability invariance, (i) stability change, (iii) stability switching, (iv) instability invariance, and (v) instability switching. Even one of these dynamics can occur with multiple mechanisms based on the distribution of critical time delays. Generally, two or three types of dynamics are detected in many population models, but exhibiting all the above dynamics is not observed. In an ecological system, species form groups to improve their chances of survival. Taking inspiration from tuna’s forging behavior Cosner et al. (1999) developed the Cosner functional response. In this study, we propose a delayed predator–prey model with Cosner functional response. The non-delayed model can have up to four equilibria, two coexisting equilibria (anti-saddle and saddle), along with trivial and boundary equilibria. The stability of all equilibria is analyzed with time delay. Under certain parameter conditions, the boundary equilibrium remains globally stable for all delays. For increasing delay, the anti-saddle equilibrium may: (i) remain stable, (ii) undergo stability change (two possible scenarios), (iii) undergo stability switching, (iv) remain unstable (two possible scenarios), or (v) undergo instability switching. These seven stability scenarios are verified to exhibit, while an additional instability invariance scenario, where no critical delay exists, is analytically shown to be non-existent. Showing all these mentioned stability scenarios in a predator–prey model with a single delay is a novelty of this paper. If the anti-saddle equilibrium is stable in the absence of delay, then the degenerate case may occur, which implies the local stability between any two consecutive delay thresholds. Moreover, we have analytically proved that the degenerate case is not possible if the anti-saddle equilibrium is unstable in the absence of delay, which is a new observation in population dynamics. We have computed species survival basin for increasing delay. Our investigation reveals that increasing delay can change the shape and size of the basin, making delay beneficial or harmful for the species’ survival, depending on the initial populations of species. Finally, we have proposed an open question and outlined a couple of potential directions for future research.
时间延迟是生态过程不可或缺的一部分。种群模型包含了时间延迟，以解释成熟、孕育、扩散等所需的时间。时间延迟可以诱发各种稳定性动力学，包括(i)稳定性不变性，(i)稳定性变化，（iii）稳定性切换，（iv）不稳定性不变性和(v)不稳定性切换。甚至这些动态中的一种也可能发生在基于临界时间延迟分布的多种机制中。通常，在许多种群模型中检测到两种或三种类型的动态，但没有观察到表现出上述所有动态。在生态系统中，物种形成群体是为了提高生存的机会。Cosner et al.（1999）从金枪鱼的锻造行为中获得灵感，开发了Cosner功能反应。在本研究中，我们提出了一个具有Cosner功能响应的延迟捕食者-猎物模型。非延迟模型最多可以有四个平衡点，两个共存平衡点（反鞍态和鞍态），以及平凡平衡点和边界平衡点。用时滞分析了所有平衡点的稳定性。在一定的参数条件下，边界平衡对所有时滞保持全局稳定。对于增加的延迟，反鞍平衡可能：(i)保持稳定，（ii）经历稳定性变化（两种可能的情况），（iii）经历稳定性切换，（iv）保持不稳定（两种可能的情况），或(v)经历不稳定切换。这七个稳定性场景经过验证，而另一个不稳定不变性场景（不存在临界延迟）分析显示不存在。在具有单一延迟的捕食者-猎物模型中显示所有上述稳定性情景是本文的一个新颖之处。如果在没有延迟的情况下，反鞍平衡是稳定的，则可能出现退化情况，这意味着任意两个连续延迟阈值之间的局部稳定性。此外，我们还解析地证明了在没有时滞的情况下，如果反鞍平衡是不稳定的，就不可能出现退化情况，这是种群动力学中的一个新的观察结果。我们计算了增加延迟的物种生存盆地。我们的研究表明，延迟的增加可以改变盆地的形状和大小，使延迟对物种的生存有利或有害，这取决于物种的初始种群。最后，我们提出了一个开放性问题，并概述了未来研究的几个潜在方向。
{"title":"Delay-induced multiple stability scenarios, species coexistence, and predator extinction in an ecological system","authors":"Yovan Singh , Bapan Ghosh , Suman Mondal","doi":"10.1016/j.matcom.2025.09.025","DOIUrl":"10.1016/j.matcom.2025.09.025","url":null,"abstract":"<div><div>Time delays are integral to ecological processes. Population models incorporate time delays to account for the time required for maturation, gestation, dispersal, and many more. Time delay can induce various stability dynamics, including (i) stability invariance, (i) stability change, (iii) stability switching, (iv) instability invariance, and (v) instability switching. Even one of these dynamics can occur with multiple mechanisms based on the distribution of critical time delays. Generally, two or three types of dynamics are detected in many population models, but exhibiting all the above dynamics is not observed. In an ecological system, species form groups to improve their chances of survival. Taking inspiration from tuna’s forging behavior Cosner et al. (1999) developed the Cosner functional response. In this study, we propose a delayed predator–prey model with Cosner functional response. The non-delayed model can have up to four equilibria, two coexisting equilibria (anti-saddle and saddle), along with trivial and boundary equilibria. The stability of all equilibria is analyzed with time delay. Under certain parameter conditions, the boundary equilibrium remains globally stable for all delays. For increasing delay, the anti-saddle equilibrium may: (i) remain stable, (ii) undergo stability change (two possible scenarios), (iii) undergo stability switching, (iv) remain unstable (two possible scenarios), or (v) undergo instability switching. These seven stability scenarios are verified to exhibit, while an additional instability invariance scenario, where no critical delay exists, is analytically shown to be non-existent. Showing all these mentioned stability scenarios in a predator–prey model with a single delay is a novelty of this paper. If the anti-saddle equilibrium is stable in the absence of delay, then the degenerate case may occur, which implies the local stability between any two consecutive delay thresholds. Moreover, we have analytically proved that the degenerate case is not possible if the anti-saddle equilibrium is unstable in the absence of delay, which is a new observation in population dynamics. We have computed species survival basin for increasing delay. Our investigation reveals that increasing delay can change the shape and size of the basin, making delay beneficial or harmful for the species’ survival, depending on the initial populations of species. Finally, we have proposed an open question and outlined a couple of potential directions for future research.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"243 ","pages":"Pages 171-195"},"PeriodicalIF":4.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work introduces a ratio-dependent Holling–Tanner predator–prey model with the Allee effect in prey and then discretizes the introduced model through the Euler forward scheme. A brief discussion is held on the stability analysis for several fixed points in the discretized model. Several types of bifurcations, including codimension one and two bifurcations, are demonstrated in this study. Codimension-1 bifurcation, which covers Neimark–Sacker and flip bifurcations, and codimension-2 bifurcations, which include strong resonance 1:2, 1:3, and 1:4 at a positive fixed point. Various critical states under non-degeneracy conditions are computed using the critical normal form coefficient approach for each bifurcation. The model displays complex dynamical behaviours, like quasi-periodic orbits and chaotic sets. Additionally, the system’s chaos was managed by the development of control mechanisms, such as the OGY methodology. It has been established that bifurcation and chaos can be stabilized under certain circumstances. A thorough numerical simulation further supports our analytical findings, which include stability regions, bifurcation curves in 2D & 3D, phase plots, and the maximal Lyapunov exponent, etc.
{"title":"Multiple bifurcations and managing chaos: A discretized ratio-dependent Holling–Tanner predator–prey model with Allee effect in prey","authors":"Md. Jasim Uddin , Savita Boora , Sarker Md. Sohel Rana , Pradeep Malik","doi":"10.1016/j.matcom.2025.11.024","DOIUrl":"10.1016/j.matcom.2025.11.024","url":null,"abstract":"<div><div>This work introduces a ratio-dependent Holling–Tanner predator–prey model with the Allee effect in prey and then discretizes the introduced model through the Euler forward scheme. A brief discussion is held on the stability analysis for several fixed points in the discretized model. Several types of bifurcations, including codimension one and two bifurcations, are demonstrated in this study. Codimension-1 bifurcation, which covers Neimark–Sacker and flip bifurcations, and codimension-2 bifurcations, which include strong resonance 1:2, 1:3, and 1:4 at a positive fixed point. Various critical states under non-degeneracy conditions are computed using the critical normal form coefficient approach for each bifurcation. The model displays complex dynamical behaviours, like quasi-periodic orbits and chaotic sets. Additionally, the system’s chaos was managed by the development of control mechanisms, such as the OGY methodology. It has been established that bifurcation and chaos can be stabilized under certain circumstances. A thorough numerical simulation further supports our analytical findings, which include stability regions, bifurcation curves in 2D & 3D, phase plots, and the maximal Lyapunov exponent, etc.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"243 ","pages":"Pages 95-120"},"PeriodicalIF":4.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This study investigates a fractional-order three-disk dynamo system incorporating time delay and viscous friction, enhancing its relevance to real-world phenomena. We analyze dynamics of the system with and without time delay, revealing richer behaviors in the delayed case. Through theoretical analysis, we investigate equilibrium points and their stability, identifying pitchfork and double-Hopf bifurcations that lead to complex dynamics, including three-dimensional torus structures. Numerical simulations validate these findings for both fractional and classical systems, highlighting the impact of fractional-order derivatives and time delays. A comparative analysis shows that the fractional-order system exhibits a broader stability region than its integer-order counterpart, underscoring the stabilizing role of fractional calculus. These results provide insights into modeling magnetic field dynamics in geophysical and astrophysical systems, with potential applications to geomagnetic reversals and stellar magnetic cycles.
{"title":"Stability and bifurcation of a time-delayed fractional three-disk system","authors":"Elham Ghafari , Reza Khoshsiar Ghaziani , Javad Alidousti , Khayyam Salehi","doi":"10.1016/j.matcom.2025.11.023","DOIUrl":"10.1016/j.matcom.2025.11.023","url":null,"abstract":"<div><div>This study investigates a fractional-order three-disk dynamo system incorporating time delay and viscous friction, enhancing its relevance to real-world phenomena. We analyze dynamics of the system with and without time delay, revealing richer behaviors in the delayed case. Through theoretical analysis, we investigate equilibrium points and their stability, identifying pitchfork and double-Hopf bifurcations that lead to complex dynamics, including three-dimensional torus structures. Numerical simulations validate these findings for both fractional and classical systems, highlighting the impact of fractional-order derivatives and time delays. A comparative analysis shows that the fractional-order system exhibits a broader stability region than its integer-order counterpart, underscoring the stabilizing role of fractional calculus. These results provide insights into modeling magnetic field dynamics in geophysical and astrophysical systems, with potential applications to geomagnetic reversals and stellar magnetic cycles.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"243 ","pages":"Pages 16-34"},"PeriodicalIF":4.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145555217","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-05-01Epub Date: 2025-12-10DOI: 10.1016/j.matcom.2025.12.006
Sandra Carillo, Costanza Conti, Daniela Mansutti, Francesca Pitolli, Rosa Maria Spitaleri
{"title":"Mathematical models, numerical methods and scientific computing technologies for new arising problems (MATHSCICOMP2023)","authors":"Sandra Carillo, Costanza Conti, Daniela Mansutti, Francesca Pitolli, Rosa Maria Spitaleri","doi":"10.1016/j.matcom.2025.12.006","DOIUrl":"10.1016/j.matcom.2025.12.006","url":null,"abstract":"","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"243 ","pages":"Pages 483-485"},"PeriodicalIF":4.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-05-01Epub Date: 2025-11-26DOI: 10.1016/j.matcom.2025.11.034
Giacomo Speroni , Nicola Ferro
We present a novel metric-based anisotropic mesh adaptation algorithm, named , to be employed for discretization of three-dimensional periodic domains. The proposed method – based on mathematically rigorous assumptions – utilizes established techniques for unconstrained anisotropic mesh adaptation and resorts to localized manipulations on the external boundary of the mesh. In particular, the scheme comprises four steps: (i) a non-periodic initial mesh adaptation, (ii) the splitting of the obtained volumetric grid into interior and exterior tessellations, (iii) minimal local operations to yield a periodic external surface, and (iv) the assembly of the final adapted grids. To demonstrate the robustness, efficacy, and flexibility of the proposed methodology, algorithm is employed in a continuous finite element setting to tackle test cases established in the literature as well as challenging scenarios that involve various periodic requirements, domain geometries, and isotropic and anisotropic metric fields. Finally, is employed in a practical use case where mesh adaptation is tightly coupled with the solution of a time-dependent partial differential equation.
{"title":"A novel metric-based anisotropic mesh adaptation algorithm for 3D periodic domains","authors":"Giacomo Speroni , Nicola Ferro","doi":"10.1016/j.matcom.2025.11.034","DOIUrl":"10.1016/j.matcom.2025.11.034","url":null,"abstract":"<div><div>We present a novel metric-based anisotropic mesh adaptation algorithm, named <span><math><mrow><mn>3</mn><mstyle><mi>DPAMA</mi></mstyle></mrow></math></span>, to be employed for discretization of three-dimensional periodic domains. The proposed method – based on mathematically rigorous assumptions – utilizes established techniques for unconstrained anisotropic mesh adaptation and resorts to localized manipulations on the external boundary of the mesh. In particular, the scheme comprises four steps: (i) a non-periodic initial mesh adaptation, (ii) the splitting of the obtained volumetric grid into interior and exterior tessellations, (iii) minimal local operations to yield a periodic external surface, and (iv) the assembly of the final adapted grids. To demonstrate the robustness, efficacy, and flexibility of the proposed methodology, <span><math><mrow><mn>3</mn><mstyle><mi>DPAMA</mi></mstyle></mrow></math></span> algorithm is employed in a continuous finite element setting to tackle test cases established in the literature as well as challenging scenarios that involve various periodic requirements, domain geometries, and isotropic and anisotropic metric fields. Finally, <span><math><mrow><mn>3</mn><mstyle><mi>DPAMA</mi></mstyle></mrow></math></span> is employed in a practical use case where mesh adaptation is tightly coupled with the solution of a time-dependent partial differential equation.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"243 ","pages":"Pages 251-269"},"PeriodicalIF":4.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-05-01Epub Date: 2025-12-01DOI: 10.1016/j.matcom.2025.11.039
Achyuta Ranjan Dutta Mohapatra, Bhupen Deka
This article discusses a skeletal discontinuous finite element method for approximating solutions of time-harmonic Maxwell’s equations with high wave numbers. The name justifies the method because the local degrees of freedom are associated with the skeleton of the mesh. These methods are also quite popularly known as the modified weak Galerkin methods. The proposed algorithm for the time-harmonic Maxwell equations is a parameter-free, non-conforming finite element method that uses discontinuous polynomials to approximate the true solution. Due to the choice of functions in these skeletal Galerkin methods, one has the flexibility of an inbuilt weak tangential continuity incorporated in the approximation space. Optimal order of convergence for the errors has been derived in and a discretely defined -norms. Numerical computations verify the theoretical convergence rates, and the proposed numerical approximation scheme is stable for the time-harmonic equations with large wave numbers.
{"title":"Convergence analysis of a skeletal discontinuous Galerkin finite element method for time-harmonic Maxwell equations with large wave numbers","authors":"Achyuta Ranjan Dutta Mohapatra, Bhupen Deka","doi":"10.1016/j.matcom.2025.11.039","DOIUrl":"10.1016/j.matcom.2025.11.039","url":null,"abstract":"<div><div>This article discusses a skeletal discontinuous finite element method for approximating solutions of time-harmonic Maxwell’s equations with high wave numbers. The name justifies the method because the local degrees of freedom are associated with the skeleton of the mesh. These methods are also quite popularly known as the modified weak Galerkin methods. The proposed algorithm for the time-harmonic Maxwell equations is a parameter-free, non-conforming finite element method that uses discontinuous polynomials to approximate the true solution. Due to the choice of functions in these skeletal Galerkin methods, one has the flexibility of an inbuilt weak tangential continuity incorporated in the approximation space. Optimal order of convergence for the errors has been derived in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and a discretely defined <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>curl</mi><mo>)</mo></mrow></mrow></math></span>-norms. Numerical computations verify the theoretical convergence rates, and the proposed numerical approximation scheme is stable for the time-harmonic equations with large wave numbers.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"243 ","pages":"Pages 407-426"},"PeriodicalIF":4.4,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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