Pub Date : 2025-01-03DOI: 10.1016/j.cam.2024.116208
Zhenwu Fu , Yang Li , Yong Chen , Bo Han , Hao Tian
In this paper, we propose an accelerated Bouligand–Landweber method which is based on projection and Nesterov acceleration. This approach incorporates Nesterov acceleration technique into the Bouligand Landweber method whose step sizes are determined by projection. It is designed to solve nonsmooth ill-posed problems and to reduce the computational time. When the data is exact, we show the convergence result of the proposed method. When the data is contaminated by noise, we prove its regularization property by utilizing the concept of asymptotic stability. Moreover, some numerical experiments on nonsmooth inverse problems are performed to demonstrate the efficiency and the acceleration effect of the method.
{"title":"An accelerated Bouligand–Landweber method based on projection and Nesterov acceleration for nonsmooth ill-posed problems","authors":"Zhenwu Fu , Yang Li , Yong Chen , Bo Han , Hao Tian","doi":"10.1016/j.cam.2024.116208","DOIUrl":"10.1016/j.cam.2024.116208","url":null,"abstract":"<div><div>In this paper, we propose an accelerated Bouligand–Landweber method which is based on projection and Nesterov acceleration. This approach incorporates Nesterov acceleration technique into the Bouligand Landweber method whose step sizes are determined by projection. It is designed to solve nonsmooth ill-posed problems and to reduce the computational time. When the data is exact, we show the convergence result of the proposed method. When the data is contaminated by noise, we prove its regularization property by utilizing the concept of asymptotic stability. Moreover, some numerical experiments on nonsmooth inverse problems are performed to demonstrate the efficiency and the acceleration effect of the method.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"462 ","pages":"Article 116208"},"PeriodicalIF":2.1,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143145547","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 : 2025-01-02DOI: 10.1016/j.cam.2024.116470
Qianqian Ding , Mingxia Li
This paper rigorously analyzes a finite element method for incompressible magnetohydrodynamics flows with variable density. A fully discrete scheme based on the Euler semi-implicit method is proposed. The magnetic equation is approximated by Ndlec edge elements, the density equation is approximated by Discontinuous Galerkin method, and the momentum equations are approximated by continuous elements. The numerical scheme is showed to satisfy the laws of mass conservation and energy conservation. In addition, we prove that the discrete density system satisfies the stability, consistency and convergence. Employing the Lax–Milgram theorem, the existence of solution to the fully discrete scheme is demonstrated. As both meshwidth and timestep size tend to zero, we prove that the fully discrete solution converges to a weak solution of the continuous problem.
{"title":"Convergence analysis of finite element method for incompressible magnetohydrodynamics system with variable density","authors":"Qianqian Ding , Mingxia Li","doi":"10.1016/j.cam.2024.116470","DOIUrl":"10.1016/j.cam.2024.116470","url":null,"abstract":"<div><div>This paper rigorously analyzes a finite element method for incompressible magnetohydrodynamics flows with variable density. A fully discrete scheme based on the Euler semi-implicit method is proposed. The magnetic equation is approximated by N<span><math><mover><mrow><mi>e</mi></mrow><mrow><mo>́</mo></mrow></mover></math></span>d<span><math><mover><mrow><mi>e</mi></mrow><mrow><mo>́</mo></mrow></mover></math></span>lec edge elements, the density equation is approximated by Discontinuous Galerkin method, and the momentum equations are approximated by continuous elements. The numerical scheme is showed to satisfy the laws of mass conservation and energy conservation. In addition, we prove that the discrete density system satisfies the stability, consistency and convergence. Employing the Lax–Milgram theorem, the existence of solution to the fully discrete scheme is demonstrated. As both meshwidth and timestep size tend to zero, we prove that the fully discrete solution converges to a weak solution of the continuous problem.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"462 ","pages":"Article 116470"},"PeriodicalIF":2.1,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143145544","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 : 2025-01-02DOI: 10.1016/j.cam.2024.116481
Xiaoli Huang , Yuelin Gao , Bo Zhang , Xia Liu
This paper proposes an outer space branch-reduction-bound algorithm by using the second-order cone relaxation technique with regional reduction strategy for solving generalized linear multiplicative programming (GLMP) problems. Firstly, by introducing additional variables and performing several equivalent transformations, the GLMP problem is converted into an equivalent problem that is easier to solve. Next, the non-convex quadratic constraint in the equivalent problem is handled utilizing the secant approximation method to formulate the second-order cone relaxation problem. Subsequently, we incorporate a regional reduction technique into the algorithm to enhance its computational performance and estimate the worst-case computational complexity, ensuring the attainment of the global optimal solution. Finally, numerical experimental results demonstrate that the proposed algorithm can efficiently search for the global -optimal solution of the test examples and can successfully solve high-dimensional GLMP problems with a small number of linear functions.
{"title":"An outer space branch-reduction-bound algorithm using second-order cone relaxation with regional reduction strategy for solving equivalent generalized linear multiplicative programming","authors":"Xiaoli Huang , Yuelin Gao , Bo Zhang , Xia Liu","doi":"10.1016/j.cam.2024.116481","DOIUrl":"10.1016/j.cam.2024.116481","url":null,"abstract":"<div><div>This paper proposes an outer space branch-reduction-bound algorithm by using the second-order cone relaxation technique with regional reduction strategy for solving generalized linear multiplicative programming (GLMP) problems. Firstly, by introducing additional variables and performing several equivalent transformations, the GLMP problem is converted into an equivalent problem that is easier to solve. Next, the non-convex quadratic constraint in the equivalent problem is handled utilizing the secant approximation method to formulate the second-order cone relaxation problem. Subsequently, we incorporate a regional reduction technique into the algorithm to enhance its computational performance and estimate the worst-case computational complexity, ensuring the attainment of the global optimal solution. Finally, numerical experimental results demonstrate that the proposed algorithm can efficiently search for the global <span><math><mi>ϵ</mi></math></span>-optimal solution of the test examples and can successfully solve high-dimensional GLMP problems with a small number of linear functions.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"462 ","pages":"Article 116481"},"PeriodicalIF":2.1,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143145548","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 : 2025-01-02DOI: 10.1016/j.cam.2024.116467
Yang Zhang, Fan Zhang
In this study, we present a novel robust discontinuous Galerkin (DG) method based on the Harten–Lax–van Leer-contact (HLLC) approximate Riemann solver for weakly compressible two-phase flows governed by a three-equation model. The proposed method satisfies the mechanical equilibrium criterion which states that uniform velocity and pressure should be remained uniform during the simulation. It also maintains a positive density solution and an oscillation-free material interface by employing a positivity-preserving limiter and a compact multi-resolution weighted essentially non-oscillatory (MRWENO) limiter without violating the mechanical equilibrium criterion. A series of one- and two-dimensional numerical results are presented to demonstrate the exceptional accuracy and robustness of the proposed method. More importantly, based on the extensive numerical results, we successfully derive a suitable choice on the linear weight of the MRWENO limiter, which plays an important role in both accuracy and robustness in simulating weakly compressible two-phase flows.
{"title":"A positivity-preserving HLLC-based discontinuous Galerkin method for weakly compressible two-phase flows","authors":"Yang Zhang, Fan Zhang","doi":"10.1016/j.cam.2024.116467","DOIUrl":"10.1016/j.cam.2024.116467","url":null,"abstract":"<div><div>In this study, we present a novel robust discontinuous Galerkin (DG) method based on the Harten–Lax–van Leer-contact (HLLC) approximate Riemann solver for weakly compressible two-phase flows governed by a three-equation model. The proposed method satisfies the mechanical equilibrium criterion which states that uniform velocity and pressure should be remained uniform during the simulation. It also maintains a positive density solution and an oscillation-free material interface by employing a positivity-preserving limiter and a compact multi-resolution weighted essentially non-oscillatory (MRWENO) limiter without violating the mechanical equilibrium criterion. A series of one- and two-dimensional numerical results are presented to demonstrate the exceptional accuracy and robustness of the proposed method. More importantly, based on the extensive numerical results, we successfully derive a suitable choice on the linear weight of the MRWENO limiter, which plays an important role in both accuracy and robustness in simulating weakly compressible two-phase flows.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"462 ","pages":"Article 116467"},"PeriodicalIF":2.1,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143145525","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 : 2025-01-02DOI: 10.1016/j.cam.2024.116473
Xuhao Li , Patricia J.Y. Wong , Anatoly A. Alikhanov
Using a new generalized formula and a time varying compact finite difference operator, we construct a high order numerical scheme for a class of generalized fractional diffusion equation with space–time varying diffusivity that admits a smooth solution. The convergence order is shown to be via the energy method and demonstrated by numerical experiments. Our contributions, which improve some previous work, focus primarily on two aspects: (i) we develop a novel generalized formula achieving accuracy; (ii) we derive an essential a priori estimate for a time-varying compact finite difference operator, ensuring the new numerical scheme is stable and convergent.
{"title":"Numerical method for fractional sub-diffusion equation with space–time varying diffusivity and smooth solution","authors":"Xuhao Li , Patricia J.Y. Wong , Anatoly A. Alikhanov","doi":"10.1016/j.cam.2024.116473","DOIUrl":"10.1016/j.cam.2024.116473","url":null,"abstract":"<div><div>Using a new generalized <span><math><mrow><mi>L</mi><mn>2</mn></mrow></math></span> formula and a time varying compact finite difference operator, we construct a high order numerical scheme for a class of generalized fractional diffusion equation with space–time varying diffusivity that admits a smooth solution. The convergence order is shown to be <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>z</mi></mrow><mrow><mn>3</mn><mo>−</mo><mi>α</mi></mrow></msubsup><mo>+</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> via the energy method and demonstrated by numerical experiments. Our contributions, which improve some previous work, focus primarily on two aspects: (i) we develop a novel generalized <span><math><mrow><mi>L</mi><mn>2</mn></mrow></math></span> formula achieving <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msubsup><mrow><mi>τ</mi></mrow><mrow><mi>z</mi></mrow><mrow><mn>3</mn><mo>−</mo><mi>α</mi></mrow></msubsup><mo>)</mo></mrow></mrow></math></span> accuracy; (ii) we derive an essential a priori estimate for a time-varying compact finite difference operator, ensuring the new numerical scheme is stable and convergent.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"464 ","pages":"Article 116473"},"PeriodicalIF":2.1,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143169623","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 : 2025-01-02DOI: 10.1016/j.cam.2024.116474
Siegfried Macías , Jorge E. Macías-Díaz
In this work, we extend the Zakharov–Rubenchik system to the fractional case by using Riesz operators of fractional order in space. We prove that the system is capable of preserving extensions of the mass, energy, momentum and two linear functionals. In a second stage, we propose a discretization to approximate the solutions of our model. In the way, we propose discrete forms of the conserved functionals, and we prove that they are also conserved in the discrete domain. We prove that the numerical scheme has second-order accuracy in both space and time. Moreover, we establish theoretically the properties of conditional stability and second-order convergence of the scheme. The numerical model was implemented computationally, and some simulations are provided in order to illustrate that the method is capable of conserving the discrete functionals and its rate of convergence. This is the first report in the literature in which a multi-conservative fractional extension of this system is proposed, and a numerical scheme to approximate its solutions is designed and fully analyzed for conservative and numerical properties. Even in the integer-order case, this is the first article which proves the approximate conservation of the momentum, and which rigorously proves the stability and the convergence of a scheme for the Zakharov–Rubenchik system.
{"title":"On a fractional generalization of a nonlinear model in plasma physics and its numerical resolution via a multi-conservative and efficient scheme","authors":"Siegfried Macías , Jorge E. Macías-Díaz","doi":"10.1016/j.cam.2024.116474","DOIUrl":"10.1016/j.cam.2024.116474","url":null,"abstract":"<div><div>In this work, we extend the Zakharov–Rubenchik system to the fractional case by using Riesz operators of fractional order in space. We prove that the system is capable of preserving extensions of the mass, energy, momentum and two linear functionals. In a second stage, we propose a discretization to approximate the solutions of our model. In the way, we propose discrete forms of the conserved functionals, and we prove that they are also conserved in the discrete domain. We prove that the numerical scheme has second-order accuracy in both space and time. Moreover, we establish theoretically the properties of conditional stability and second-order convergence of the scheme. The numerical model was implemented computationally, and some simulations are provided in order to illustrate that the method is capable of conserving the discrete functionals and its rate of convergence. This is the first report in the literature in which a multi-conservative fractional extension of this system is proposed, and a numerical scheme to approximate its solutions is designed and fully analyzed for conservative and numerical properties. Even in the integer-order case, this is the first article which proves the approximate conservation of the momentum, and which rigorously proves the stability and the convergence of a scheme for the Zakharov–Rubenchik system.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"462 ","pages":"Article 116474"},"PeriodicalIF":2.1,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143145539","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 : 2025-01-01DOI: 10.1016/j.cam.2024.116469
Danxia Wang , Xuliang Wei , Chuanjun Chen
In this paper, we study the numerical approximation of the Allen–Cahn(AC) equation. Firstly, based on the Lagrange multiplier strategy, we present an equivalent form of the AC equation. Secondly, we provide corresponding semi-discrete and fully-discrete finite element schemes which satisfy the energy law of dissipation. Thirdly, we derive optimal error estimates of the semi-discrete and fully-discrete schemes. That is, by eliminating the Lagrange multiplier of the above discrete schemes, we obtain equivalent equations with only the phase field variable, then we establish the bounds and bounds of the numerical solution using mathematical induction. Due to the spatial discretization of the fully-discrete scheme, we derive more necessary conclusions to make preparation for the optimal error estimation. With the use of the spectrum argument, the superconvergence property of nonlinear terms and the mathematical induction, we obtain error bounds that solely depend on low-order polynomial of rather than the exponential factor . Finally, we present some numerical experiments to validate our theoretical convergence analysis.
{"title":"Optimal error estimates for the semi-discrete and fully-discrete finite element schemes of the Allen–Cahn equation","authors":"Danxia Wang , Xuliang Wei , Chuanjun Chen","doi":"10.1016/j.cam.2024.116469","DOIUrl":"10.1016/j.cam.2024.116469","url":null,"abstract":"<div><div>In this paper, we study the numerical approximation of the Allen–Cahn(AC) equation. Firstly, based on the Lagrange multiplier strategy, we present an equivalent form of the AC equation. Secondly, we provide corresponding semi-discrete and fully-discrete finite element schemes which satisfy the energy law of dissipation. Thirdly, we derive optimal error estimates of the semi-discrete and fully-discrete schemes. That is, by eliminating the Lagrange multiplier of the above discrete schemes, we obtain equivalent equations with only the phase field variable, then we establish the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span> bounds and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> bounds of the numerical solution using mathematical induction. Due to the spatial discretization of the fully-discrete scheme, we derive more necessary conclusions to make preparation for the optimal error estimation. With the use of the spectrum argument, the superconvergence property of nonlinear terms and the mathematical induction, we obtain error bounds that solely depend on low-order polynomial of <span><math><msup><mrow><mi>ɛ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> rather than the exponential factor <span><math><msup><mrow><mi>e</mi></mrow><mrow><mfrac><mrow><mi>T</mi></mrow><mrow><msup><mrow><mi>ɛ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></msup></math></span>. Finally, we present some numerical experiments to validate our theoretical convergence analysis.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"462 ","pages":"Article 116469"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143145546","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 : 2025-01-01DOI: 10.1016/j.cam.2024.116468
Wei Liu, Pengshan Wang, Gexian Fan
In this paper, Brinkman flow is introduced to simulate fluid flow within fractures affected by the resistance from porous media and viscous shear on fractures walls. Due to the existence of chemical reactions, reactive transport is considered in fractured porous media. The reactions alter the porous media and fractures locally, thus the equations for evolution of porosity, fracture aperture and precipitation concentration are extended to the highly coupled model of Brinkman flow and advection-diffusion-reaction transport in fractured porous media. Generally, fractures and their intersections are treated as low-dimensional immersed objects to obtain hybrid-dimensional coupled models. A space–time algorithm based on finite volume method is constructed to solve the hybrid-dimensional coupled system, by decoupling the ODEs and PDEs at each time level sequentially. The simulation of each physical process including discontinuous pressure, concentration and reaction terms can be realized effectively. Besides, the proposed algorithm can be developed to high-dimensional porous media with multiple intersecting fractures easily. Error estimates illustrate that the proposed algorithm can achieve space–time second-order accuracy. Numerical experiments are provided to confirm the accuracy and effectiveness of the proposed algorithm with variable temporal steps in porous media embedded with variable fractures.
{"title":"A space–time second-order algorithm based on finite volume method for Brinkman flow and reactive transport model in porous media with variable fractures","authors":"Wei Liu, Pengshan Wang, Gexian Fan","doi":"10.1016/j.cam.2024.116468","DOIUrl":"10.1016/j.cam.2024.116468","url":null,"abstract":"<div><div>In this paper, Brinkman flow is introduced to simulate fluid flow within fractures affected by the resistance from porous media and viscous shear on fractures walls. Due to the existence of chemical reactions, reactive transport is considered in fractured porous media. The reactions alter the porous media and fractures locally, thus the equations for evolution of porosity, fracture aperture and precipitation concentration are extended to the highly coupled model of Brinkman flow and advection-diffusion-reaction transport in fractured porous media. Generally, fractures and their intersections are treated as low-dimensional immersed objects to obtain hybrid-dimensional coupled models. A space–time algorithm based on finite volume method is constructed to solve the hybrid-dimensional coupled system, by decoupling the ODEs and PDEs at each time level sequentially. The simulation of each physical process including discontinuous pressure, concentration and reaction terms can be realized effectively. Besides, the proposed algorithm can be developed to high-dimensional porous media with multiple intersecting fractures easily. Error estimates illustrate that the proposed algorithm can achieve space–time second-order accuracy. Numerical experiments are provided to confirm the accuracy and effectiveness of the proposed algorithm with variable temporal steps in porous media embedded with variable fractures.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"462 ","pages":"Article 116468"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143145538","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 : 2025-01-01DOI: 10.1016/j.cam.2024.116465
Mohammed Elghandouri , Khalil Ezzinbi , Lamiae Saidi
In this paper, the existence, uniqueness, and positivity of solutions, as well as the asymptotic behavior through a finite fractal dimensional global attractor for a general Advection–Diffusion–Reaction (ADR) equation, are investigated. Our findings are innovative, as we employ semigroups and global attractors theories to achieve these results. Also, an analytical solution of a two-dimensional Advection–Diffusion Equation is presented. And finally, two Explicit Finite Difference schemes are used to simulate solutions in the two- and three-dimensional cases. The numerical simulations are conducted with predefined initial and Dirichlet boundary conditions.
{"title":"Exploring well-posedness and asymptotic behavior in an Advection–Diffusion–Reaction (ADR) model","authors":"Mohammed Elghandouri , Khalil Ezzinbi , Lamiae Saidi","doi":"10.1016/j.cam.2024.116465","DOIUrl":"10.1016/j.cam.2024.116465","url":null,"abstract":"<div><div>In this paper, the existence, uniqueness, and positivity of solutions, as well as the asymptotic behavior through a finite fractal dimensional global attractor for a general Advection–Diffusion–Reaction (ADR) equation, are investigated. Our findings are innovative, as we employ semigroups and global attractors theories to achieve these results. Also, an analytical solution of a two-dimensional Advection–Diffusion Equation is presented. And finally, two Explicit Finite Difference schemes are used to simulate solutions in the two- and three-dimensional cases. The numerical simulations are conducted with predefined initial and Dirichlet boundary conditions.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"462 ","pages":"Article 116465"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143145541","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 : 2025-01-01DOI: 10.1016/j.cam.2024.116466
Huihui Cao , Hengguang Li , Nianyu Yi , Peimeng Yin
In this paper, we consider an adaptive finite element method for solving elliptic equations with line Dirac delta functions as the source term. Instead of using a local local indicator, or regularizing the singular source term and using the classical residual-based a posteriori error estimator, we propose a novel a posteriori estimator based on an equivalent transmission problem. This equivalent problem is defined in the same domain as the original problem but features a zero source term and nonzero flux jumps along the line cracks, leading to a more regular solution. The a posteriori error estimator relies on meshes that conform to the line cracks, and its edge jump residual essentially incorporates the flux jumps of the transmission problem on these cracks. The proposed error estimator is proven to be both reliable and efficient. We also introduce an adaptive finite element algorithm based on this error estimator and the bisection refinement method. Numerical tests demonstrate that quasi-optimal convergence rates are achieved for both low-order and high-order approximations, with the associated adaptive meshes primarily refined at a finite number of singular points in the domain.
{"title":"Regularity and an adaptive finite element method for elliptic equations with Dirac sources on line cracks","authors":"Huihui Cao , Hengguang Li , Nianyu Yi , Peimeng Yin","doi":"10.1016/j.cam.2024.116466","DOIUrl":"10.1016/j.cam.2024.116466","url":null,"abstract":"<div><div>In this paper, we consider an adaptive finite element method for solving elliptic equations with line Dirac delta functions as the source term. Instead of using a local <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> local indicator, or regularizing the singular source term and using the classical residual-based a posteriori error estimator, we propose a novel a posteriori estimator based on an equivalent transmission problem. This equivalent problem is defined in the same domain as the original problem but features a zero source term and nonzero flux jumps along the line cracks, leading to a more regular solution. The a posteriori error estimator relies on meshes that conform to the line cracks, and its edge jump residual essentially incorporates the flux jumps of the transmission problem on these cracks. The proposed error estimator is proven to be both reliable and efficient. We also introduce an adaptive finite element algorithm based on this error estimator and the bisection refinement method. Numerical tests demonstrate that quasi-optimal convergence rates are achieved for both low-order and high-order approximations, with the associated adaptive meshes primarily refined at a finite number of singular points in the domain.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"462 ","pages":"Article 116466"},"PeriodicalIF":2.1,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143145537","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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