Pub Date : 2024-11-01DOI: 10.1016/j.rinam.2024.100511
Jay Gopalakrishnan , Michael Neunteufel , Joachim Schöberl , Max Wardetzky
Although Regge finite element functions are not continuous, useful generalizations of nonlinear derivatives like the curvature, can be defined using them. This paper is devoted to studying the convergence of the finite element lifting of a generalized (distributional) Gauss curvature defined using a metric tensor approximation in the Regge finite element space. Specifically, we investigate the interplay between the polynomial degree of the curvature lifting by Lagrange elements and the degree of the metric tensor in the Regge finite element space. Previously, a superconvergence result, where convergence rate of one order higher than expected, was obtained when the approximate metric is the canonical Regge interpolant of the exact metric. In this work, we show that an even higher order can be obtained if the degree of the curvature lifting is reduced by one polynomial degree and if at least linear Regge elements are used. These improved convergence rates are confirmed by numerical examples.
{"title":"On the improved convergence of lifted distributional Gauss curvature from Regge elements","authors":"Jay Gopalakrishnan , Michael Neunteufel , Joachim Schöberl , Max Wardetzky","doi":"10.1016/j.rinam.2024.100511","DOIUrl":"10.1016/j.rinam.2024.100511","url":null,"abstract":"<div><div>Although Regge finite element functions are not continuous, useful generalizations of nonlinear derivatives like the curvature, can be defined using them. This paper is devoted to studying the convergence of the finite element lifting of a generalized (distributional) Gauss curvature defined using a metric tensor approximation in the Regge finite element space. Specifically, we investigate the interplay between the polynomial degree of the curvature lifting by Lagrange elements and the degree of the metric tensor in the Regge finite element space. Previously, a superconvergence result, where convergence rate of one order higher than expected, was obtained when the approximate metric is the canonical Regge interpolant of the exact metric. In this work, we show that an even higher order can be obtained if the degree of the curvature lifting is reduced by one polynomial degree and if at least linear Regge elements are used. These improved convergence rates are confirmed by numerical examples.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100511"},"PeriodicalIF":1.4,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142579067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-23DOI: 10.1016/j.rinam.2024.100490
Abdelhamid Mohammed Djaouti , Mourad Kainane mezadek , Mohamed Kainane mezadek , Ali M.A. Bany Awad
This paper focuses on the study of global existence (in time) of solutions to a weakly coupled system of Cauchy problem for semilinear -evolution models with -visco-elastic damping. The system consists of two equations, one involving the function and the other involving the function . The equations are characterized by a classical power nonlinearity and a derivative-type nonlinearity. The main objective is to investigate the relationship between the regularity assumptions on the initial data and the range of permissible exponents and in the power nonlinearity. The paper considers the system in a spatial domain and a time domain , with specific conditions on the parameters , , , and , under the symmetry property as well as the exponents and . The initial data are required to satisfy certain conditions in terms of their integrability and Sobolev regularity.
{"title":"Weakly coupled system of semilinear structural σ-evolution models with δ- visco-elastic damping","authors":"Abdelhamid Mohammed Djaouti , Mourad Kainane mezadek , Mohamed Kainane mezadek , Ali M.A. Bany Awad","doi":"10.1016/j.rinam.2024.100490","DOIUrl":"10.1016/j.rinam.2024.100490","url":null,"abstract":"<div><div>This paper focuses on the study of global existence (in time) of solutions to a weakly coupled system of Cauchy problem for semilinear <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-evolution models with <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-visco-elastic damping. The system consists of two equations, one involving the function <span><math><mi>u</mi></math></span> and the other involving the function <span><math><mi>v</mi></math></span>. The equations are characterized by a classical power nonlinearity and a derivative-type nonlinearity. The main objective is to investigate the relationship between the regularity assumptions on the initial data and the range of permissible exponents <span><math><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> in the power nonlinearity. The paper considers the system in a spatial domain <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> and a time domain <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></math></span>, with specific conditions on the parameters <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, and <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, under the symmetry property as well as the exponents <span><math><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. The initial data <span><math><mrow><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow></math></span> are required to satisfy certain conditions in terms of their integrability and Sobolev regularity.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100490"},"PeriodicalIF":1.4,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-22DOI: 10.1016/j.rinam.2024.100505
Abdelmajid Ezzine, Abdellah Alla, Nadia Raissi
The aim of this paper is to investigate an efficient spectral method for pricing European call options under regime-switching models. The main characteristic of this model is to incorporate the change in behavior of the underlying assets depending on different market states. The option pricing problem is modeled as a system of coupled Black–Scholes PDEs. The spatial discretization of the problem is performed using the Legendre–Galerkin spectral method based on Fourier-like basis functions, while the temporal discretization is based on a Crank–Nicolson scheme. Furthermore, the stability and convergence analysis are carried out for both the semi-and fully discretization of the resulted coupled PDE system. Finally, numerical experiments are illustrated to demonstrate the practical application potential of the discussed approach and its efficiency in real world cases.
{"title":"A Legendre–Galerkin spectral method for option pricing under regime switching models","authors":"Abdelmajid Ezzine, Abdellah Alla, Nadia Raissi","doi":"10.1016/j.rinam.2024.100505","DOIUrl":"10.1016/j.rinam.2024.100505","url":null,"abstract":"<div><div>The aim of this paper is to investigate an efficient spectral method for pricing European call options under regime-switching models. The main characteristic of this model is to incorporate the change in behavior of the underlying assets depending on different market states. The option pricing problem is modeled as a system of coupled Black–Scholes PDEs. The spatial discretization of the problem is performed using the Legendre–Galerkin spectral method based on Fourier-like basis functions, while the temporal discretization is based on a Crank–Nicolson scheme. Furthermore, the stability and convergence analysis are carried out for both the semi-and fully discretization of the resulted coupled PDE system. Finally, numerical experiments are illustrated to demonstrate the practical application potential of the discussed approach and its efficiency in real world cases.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100505"},"PeriodicalIF":1.4,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-07DOI: 10.1016/j.rinam.2024.100502
H. Egger , S. Kurz , R. Löscher
We study damped wave propagation problems phrased as abstract evolution equations in Hilbert spaces. Under some general assumptions, including a natural compatibility condition for initial values, we establish exponential decay estimates for mild solutions using Lyapunov-type arguments. For the formulation of our results, we use the language of Hilbert complexes which provides all the tools required for our analysis and is also general enough to cover a number of interesting examples. Some of these are briefly discussed in the course of the manuscript. The functional analytic setting and the main arguments in our proofs are chosen such that they transfer almost verbatim to the discrete setting. We thus obtain corresponding decay results for numerical approximations of a variety of problems obtained by compatible discretization strategies which can be seen as our main contribution.
{"title":"On the exponential stability of uniformly damped wave equations and their structure-preserving discretization","authors":"H. Egger , S. Kurz , R. Löscher","doi":"10.1016/j.rinam.2024.100502","DOIUrl":"10.1016/j.rinam.2024.100502","url":null,"abstract":"<div><div>We study damped wave propagation problems phrased as abstract evolution equations in Hilbert spaces. Under some general assumptions, including a natural compatibility condition for initial values, we establish exponential decay estimates for mild solutions using Lyapunov-type arguments. For the formulation of our results, we use the language of Hilbert complexes which provides all the tools required for our analysis and is also general enough to cover a number of interesting examples. Some of these are briefly discussed in the course of the manuscript. The functional analytic setting and the main arguments in our proofs are chosen such that they transfer almost verbatim to the discrete setting. We thus obtain corresponding decay results for numerical approximations of a variety of problems obtained by compatible discretization strategies which can be seen as our main contribution.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100502"},"PeriodicalIF":1.4,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142420717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/j.rinam.2024.100501
Owen Tamin , Samsul Ariffin Abdul Karim , Mohammad Khatim Hasan
Scattered data interpolation is important in sciences, engineering, and medical-based problems. The degree smoothness attained is either or . However, based on our simulations, we found that, Hussain and Hussain (2011) scheme is not producing everywhere. Motivated by this, in this study we will propose an improve sufficient condition for the on each adjacent triangle. Our scheme is guaranteed to produce everywhere. To support our claim, we have implemented both schemes and perform some numerical comparison including error measurement. From the results, our proposed scheme is producing interpolating surface while the interpolating surface produced by Hussain and Hussain (2011) is not everywhere. Furthermore, our proposed scheme give smaller error compared with Hussain and Hussain (2011) scheme. All numerical and graphical results are presented by using MATLAB programming.
{"title":"An Improved C1 rational Bernstein–Bézier triangular patch for scattered data interpolation","authors":"Owen Tamin , Samsul Ariffin Abdul Karim , Mohammad Khatim Hasan","doi":"10.1016/j.rinam.2024.100501","DOIUrl":"10.1016/j.rinam.2024.100501","url":null,"abstract":"<div><div>Scattered data interpolation is important in sciences, engineering, and medical-based problems. The degree smoothness attained is either <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> or <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. However, based on our simulations, we found that, Hussain and Hussain (2011) scheme is not producing <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> everywhere. Motivated by this, in this study we will propose an improve sufficient condition for the <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> on each adjacent triangle. Our scheme is guaranteed to produce <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> everywhere. To support our claim, we have implemented both schemes and perform some numerical comparison including error measurement. From the results, our proposed scheme is producing <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> interpolating surface while the interpolating surface produced by Hussain and Hussain (2011) is not <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> everywhere. Furthermore, our proposed scheme give smaller error compared with Hussain and Hussain (2011) scheme. All numerical and graphical results are presented by using MATLAB programming.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100501"},"PeriodicalIF":1.4,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142420716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-28DOI: 10.1016/j.rinam.2024.100498
Penghui Lv , Yuxiao Cun , Guoguang Lin
This paper delves into the stochastic asymptotic behavior of a non-autonomous stochastic higher-order Kirchhoff equation with variable coefficient rotational inertia. The equation is solved using the Galerkin method, and a stochastic dynamical system is established on this basis. Uniform estimation demonstrates a family of absorbing sets in the stochastic dynamical system , and the asymptotic compactness of is proved via decomposition. Finally, the family of random attractors is acquired for the stochastic dynamical system in . These results improve and extend those in recent literature (Lv et al., 2021). The findings promote the relevant conclusions of the non-autonomous stochastic higher-order Kirchhoff model and provide a theoretical basis for its subsequent application and research.
{"title":"Long-time dynamics of a random higher-order Kirchhoff model with variable coefficient rotational inertia","authors":"Penghui Lv , Yuxiao Cun , Guoguang Lin","doi":"10.1016/j.rinam.2024.100498","DOIUrl":"10.1016/j.rinam.2024.100498","url":null,"abstract":"<div><div>This paper delves into the stochastic asymptotic behavior of a non-autonomous stochastic higher-order Kirchhoff equation with variable coefficient rotational inertia. The equation is solved using the Galerkin method, and a stochastic dynamical system is established on this basis. Uniform estimation demonstrates a family of <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>−</mo></mrow></math></span>absorbing sets in the stochastic dynamical system <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, and the asymptotic compactness of <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is proved via decomposition. Finally, the family of <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>−</mo></mrow></math></span>random attractors is acquired for the stochastic dynamical system <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> in <span><math><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>×</mo><msubsup><mrow><mi>V</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>k</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>. These results improve and extend those in recent literature (Lv et al., 2021). The findings promote the relevant conclusions of the non-autonomous stochastic higher-order Kirchhoff model and provide a theoretical basis for its subsequent application and research.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100498"},"PeriodicalIF":1.4,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142357101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-27DOI: 10.1016/j.rinam.2024.100499
Yanfang Xue, Wenjing Gu, Jianxin Han
This paper is concerned with the existence of bound state solutions for quasilinear Schrödinger equations with Hardy potential and Berestycki–Lions type conditions. By using the variational methods, we get the existence results which is a complement to the ones in Hu et al. (2022).
本文主要研究具有哈迪势和贝里斯基-狮子型条件的准线性薛定谔方程的边界解的存在性。通过使用变分法，我们得到了存在性结果，这是对 Hu 等人 (2022) 中存在性结果的补充。
{"title":"Bound state solutions for quasilinear Schrödinger equations with Hardy potential","authors":"Yanfang Xue, Wenjing Gu, Jianxin Han","doi":"10.1016/j.rinam.2024.100499","DOIUrl":"10.1016/j.rinam.2024.100499","url":null,"abstract":"<div><div>This paper is concerned with the existence of bound state solutions for quasilinear Schrödinger equations with Hardy potential and Berestycki–Lions type conditions. By using the variational methods, we get the existence results which is a complement to the ones in Hu et al. (2022).</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100499"},"PeriodicalIF":1.4,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142327941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-27DOI: 10.1016/j.rinam.2024.100500
Hyesun Na, Eunjung Lee
To investigate the discrete field continuity at the interface of inhomogeneous materials, this paper investigates the scattering of electromagnetic waves interacting with a perfect electrical conductor object coated with dielectric materials, employing the finite element method. In the derivation process of the variational formulation, the tangential continuity of electric fields eliminates any discrepancies occurring along the internal interfaces. However, while it is important to maintain tangential continuity of electric and magnetic fields at the interface between different dielectrics, this requirement is not met precisely within discrete space. As a result, approximations can exhibit inaccuracies and potentially fail to fully capture the underlying physical phenomena. To alleviate these issues, we propose an imposition of tangential continuity of the magnetic field within the minimizing functional, thereby ensuring adherence to interface conditions between two dielectric layers. This approach can be naturally applied to a variety of interface problems to enhance the approximation accuracy of interaction models in real-world environments.
{"title":"Enforcing interface field continuity to advance finite element approximations in heterogeneous materials","authors":"Hyesun Na, Eunjung Lee","doi":"10.1016/j.rinam.2024.100500","DOIUrl":"10.1016/j.rinam.2024.100500","url":null,"abstract":"<div><div>To investigate the discrete field continuity at the interface of inhomogeneous materials, this paper investigates the scattering of electromagnetic waves interacting with a perfect electrical conductor object coated with dielectric materials, employing the finite element method. In the derivation process of the variational formulation, the tangential continuity of electric fields eliminates any discrepancies occurring along the internal interfaces. However, while it is important to maintain tangential continuity of electric and magnetic fields at the interface between different dielectrics, this requirement is not met precisely within discrete space. As a result, approximations can exhibit inaccuracies and potentially fail to fully capture the underlying physical phenomena. To alleviate these issues, we propose an imposition of tangential continuity of the magnetic field within the minimizing functional, thereby ensuring adherence to interface conditions between two dielectric layers. This approach can be naturally applied to a variety of interface problems to enhance the approximation accuracy of interaction models in real-world environments.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"24 ","pages":"Article 100500"},"PeriodicalIF":1.4,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142327940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-01DOI: 10.1016/j.rinam.2024.100494
Long Chen , Xuehai Huang
A unified construction of -conforming finite element tensors, including vector element, symmetric matrix element, traceless matrix element, and, in general, tensors with linear constraints, is developed in this work. It is based on the geometric decomposition of Lagrange elements into bubble functions on each sub-simplex. Each tensor at a sub-simplex is further decomposed into tangential and normal components. The tangential component forms the bubble function space, while the normal component characterizes the trace. Some degrees of freedom can be redistributed to -dimensional faces. The developed finite element spaces are -conforming and satisfy the discrete inf-sup condition. Intrinsic bases of the constraint tensor space are also established.
{"title":"H(div)-conforming finite element tensors with constraints","authors":"Long Chen , Xuehai Huang","doi":"10.1016/j.rinam.2024.100494","DOIUrl":"10.1016/j.rinam.2024.100494","url":null,"abstract":"<div><p>A unified construction of <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mo>div</mo><mo>)</mo></mrow></mrow></math></span>-conforming finite element tensors, including vector element, symmetric matrix element, traceless matrix element, and, in general, tensors with linear constraints, is developed in this work. It is based on the geometric decomposition of Lagrange elements into bubble functions on each sub-simplex. Each tensor at a sub-simplex is further decomposed into tangential and normal components. The tangential component forms the bubble function space, while the normal component characterizes the trace. Some degrees of freedom can be redistributed to <span><math><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-dimensional faces. The developed finite element spaces are <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mo>div</mo><mo>)</mo></mrow></mrow></math></span>-conforming and satisfy the discrete inf-sup condition. Intrinsic bases of the constraint tensor space are also established.</p></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"23 ","pages":"Article 100494"},"PeriodicalIF":1.4,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2590037424000645/pdfft?md5=8716e43d076745ec3408e0af1d9f2173&pid=1-s2.0-S2590037424000645-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142240774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-01DOI: 10.1016/j.rinam.2024.100484
Hidekazu Yoshioka
We formulate an optimal control problem of resource extraction, where a decision maker with sustainability concern dynamically controls the extraction rate. We assume harvesting to increase profit and incur a risk of resource depletion and aim to resolve sustainability concerns. The optimality equation of the control problem is the Hamilton–Jacobi–Bellman (HJB) equation with an unbounded Hamiltonian. A regularization technique to bound the Hamiltonian is proposed to prove the existence of a unique viscosity solution to both the modified and original HJB equations. We also investigate a relaxed control case, an exploratory control counterpart of our mathematical model, with the control variable belonging to a set of probability measures. Convergent, fully implicit finite difference methods to compute the viscosity solutions to the HJB equations are presented as well. These numerical methods exploit the characteristic direction of the Hamiltonians to avoid using any matrix inversions. Finally, a demonstrative application example of the proposed model to a fishery management problem is presented.
{"title":"Regular and exploratory resource extraction models considering sustainability","authors":"Hidekazu Yoshioka","doi":"10.1016/j.rinam.2024.100484","DOIUrl":"10.1016/j.rinam.2024.100484","url":null,"abstract":"<div><p>We formulate an optimal control problem of resource extraction, where a decision maker with sustainability concern dynamically controls the extraction rate. We assume harvesting to increase profit and incur a risk of resource depletion and aim to resolve sustainability concerns. The optimality equation of the control problem is the Hamilton–Jacobi–Bellman (HJB) equation with an unbounded Hamiltonian. A regularization technique to bound the Hamiltonian is proposed to prove the existence of a unique viscosity solution to both the modified and original HJB equations. We also investigate a relaxed control case, an exploratory control counterpart of our mathematical model, with the control variable belonging to a set of probability measures. Convergent, fully implicit finite difference methods to compute the viscosity solutions to the HJB equations are presented as well. These numerical methods exploit the characteristic direction of the Hamiltonians to avoid using any matrix inversions. Finally, a demonstrative application example of the proposed model to a fishery management problem is presented.</p></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"23 ","pages":"Article 100484"},"PeriodicalIF":1.4,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2590037424000542/pdfft?md5=b0edaf2877c28d46961498b04a0a4236&pid=1-s2.0-S2590037424000542-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141950857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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