Pub Date : 2025-02-24DOI: 10.1016/j.camwa.2025.02.014
Federico Gatti , Carlo de Falco , Marco Fois , Luca Formaggia
We propose a scalable well-balanced numerical method to efficiently solve a modified set of shallow water equations targeting the dynamics of lava flows. The governing equations are an extension of a depth-integrated model already available in the literature and proposed to model lava flows. Here, we consider the presence of vents that act as point sources in the mass and energy equations. Starting from a scheme developed in the framework of landslide simulation, we prove its capability to deal with lava flows. We show its excellent performances in terms of parallel scaling efficiency while maintaining good results in terms of accuracy. To verify the reliability of the proposed simulation tool, we first assess the accuracy and efficiency of the scheme on ideal scenarios. In particular, we investigate the well-balancing property, we simulate benchmarks taken from the literature in the framework of lava flow simulations, and provide relevant scaling results for the parallel implementation of the method. Successively, we challenge the scheme on a real configuration taken from the available literature.
{"title":"A scalable well-balanced Taylor-Galerkin scheme for a lava flow depth-integrated model with point source vents","authors":"Federico Gatti , Carlo de Falco , Marco Fois , Luca Formaggia","doi":"10.1016/j.camwa.2025.02.014","DOIUrl":"10.1016/j.camwa.2025.02.014","url":null,"abstract":"<div><div>We propose a scalable well-balanced numerical method to efficiently solve a modified set of shallow water equations targeting the dynamics of lava flows. The governing equations are an extension of a depth-integrated model already available in the literature and proposed to model lava flows. Here, we consider the presence of vents that act as point sources in the mass and energy equations. Starting from a scheme developed in the framework of landslide simulation, we prove its capability to deal with lava flows. We show its excellent performances in terms of parallel scaling efficiency while maintaining good results in terms of accuracy. To verify the reliability of the proposed simulation tool, we first assess the accuracy and efficiency of the scheme on ideal scenarios. In particular, we investigate the well-balancing property, we simulate benchmarks taken from the literature in the framework of lava flow simulations, and provide relevant scaling results for the parallel implementation of the method. Successively, we challenge the scheme on a real configuration taken from the available literature.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"184 ","pages":"Pages 153-167"},"PeriodicalIF":2.9,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143479139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-24DOI: 10.1016/j.camwa.2025.02.019
Ömer Oruç
In the current study, we propose an accurate numerical method for plane elastostatic equations of anisotropic functionally graded materials. The proposed method uses radial basis functions augmented with polynomial basis functions in a collocation framework by employing ghost point centers which cover physical domain of considered problem. Unlike in classical collocation approach where the centers and collocation points are taken identically, using ghost centers different from the collocation points greatly improves the accuracy of the proposed method. Addition of polynomial basis function to the radial basis functions stabilized the method against shape parameter of radial basis functions and also increases accuracy of solution, mostly. Some numerical examples are solved via the proposed method both on regular and irregular domains. , and RMS error norms are calculated and for sufficient number of collocation points their values are smaller than . The obtained error norms and their comparison with other methods available in literature confirm precision of the suggested numerical method.
{"title":"The use of polynomial-augmented RBF collocation method with ghost points for plane elastostatic equations of anisotropic functionally graded materials","authors":"Ömer Oruç","doi":"10.1016/j.camwa.2025.02.019","DOIUrl":"10.1016/j.camwa.2025.02.019","url":null,"abstract":"<div><div>In the current study, we propose an accurate numerical method for plane elastostatic equations of anisotropic functionally graded materials. The proposed method uses radial basis functions augmented with polynomial basis functions in a collocation framework by employing ghost point centers which cover physical domain of considered problem. Unlike in classical collocation approach where the centers and collocation points are taken identically, using ghost centers different from the collocation points greatly improves the accuracy of the proposed method. Addition of polynomial basis function to the radial basis functions stabilized the method against shape parameter of radial basis functions and also increases accuracy of solution, mostly. Some numerical examples are solved via the proposed method both on regular and irregular domains. <span><math><msub><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>, <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and RMS error norms are calculated and for sufficient number of collocation points their values are smaller than <span><math><mn>1</mn><mi>e</mi><mo>−</mo><mn>10</mn></math></span>. The obtained error norms and their comparison with other methods available in literature confirm precision of the suggested numerical method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"184 ","pages":"Pages 116-133"},"PeriodicalIF":2.9,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143479133","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 : 2025-02-24DOI: 10.1016/j.camwa.2025.02.005
Hailong Wang , Liang Wang , Guoqing Zhu , Chunguang Xiong
We develop a novel mixed method for addressing two-dimensional Laplacian problem with Dirichlet boundary conditions, which is recast as a rot-div system of three first-order equations. We have established the well-posedness of this new method and presented the a priori error estimates. The numerical applications of Bercovier-Engelman and Ruas test cases are developed, assessing the effectiveness of the proposed rot-div mixed method. Additionally, the efficiency of the proposed mixed method is demonstrated for typical finite elements, testing the optimal convergence rate and comparing the results with analytical solutions for all unknowns and the rotation and divergence of u. Our mixed method easily generalizes to electric and magnetic boundary conditions, and mixed boundary conditions.
{"title":"Rot-div mixed finite element method of two dimensional Hodge Laplacian problem","authors":"Hailong Wang , Liang Wang , Guoqing Zhu , Chunguang Xiong","doi":"10.1016/j.camwa.2025.02.005","DOIUrl":"10.1016/j.camwa.2025.02.005","url":null,"abstract":"<div><div>We develop a novel mixed method for addressing two-dimensional Laplacian problem with Dirichlet boundary conditions, which is recast as a rot-div system of three first-order equations. We have established the well-posedness of this new method and presented the a priori error estimates. The numerical applications of Bercovier-Engelman and Ruas test cases are developed, assessing the effectiveness of the proposed rot-div mixed method. Additionally, the efficiency of the proposed mixed method is demonstrated for typical finite elements, testing the optimal convergence rate and comparing the results with analytical solutions for all unknowns and the rotation and divergence of <strong><em>u</em></strong>. Our mixed method easily generalizes to electric and magnetic boundary conditions, and mixed boundary conditions.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"184 ","pages":"Pages 134-152"},"PeriodicalIF":2.9,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143479134","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 : 2025-02-24DOI: 10.1016/j.camwa.2025.02.013
Jipei Chen , Victor M. Calo , Quanling Deng
The recently proposed soft finite element method (SoftFEM) reduces the stiffness (condition numbers), consequently improving the overall approximation accuracy. The method subtracts a least-square term that penalizes the gradient jumps across mesh interfaces from the FEM stiffness bilinear form while maintaining the system's coercivity. Herein, we present two generalizations for SoftFEM that aim to improve the approximation accuracy and further reduce the discrete systems' stiffness. Firstly and most naturally, we generalize SoftFEM by adding a least-square term to the mass bilinear form. Superconvergent results of rates and for eigenvalues are established for linear uniform elements; is the highest order of convergence known in the literature. Secondly, we generalize SoftFEM by applying the blended Gaussian-type quadratures. We demonstrate further reductions in stiffness compared to traditional FEM and SoftFEM. The coercivity and analysis of the optimal error convergences follow the work of SoftFEM. Thus, this paper focuses on the numerical study of these generalizations. For linear and uniform elements, analytical eigenpairs, exact eigenvalue errors, and superconvergent error analysis are established. Various numerical examples demonstrate the potential of generalized SoftFEMs for spectral approximation, particularly in high-frequency regimes.
{"title":"Generalized soft finite element method for elliptic eigenvalue problems","authors":"Jipei Chen , Victor M. Calo , Quanling Deng","doi":"10.1016/j.camwa.2025.02.013","DOIUrl":"10.1016/j.camwa.2025.02.013","url":null,"abstract":"<div><div>The recently proposed soft finite element method (SoftFEM) reduces the stiffness (condition numbers), consequently improving the overall approximation accuracy. The method subtracts a least-square term that penalizes the gradient jumps across mesh interfaces from the FEM stiffness bilinear form while maintaining the system's coercivity. Herein, we present two generalizations for SoftFEM that aim to improve the approximation accuracy and further reduce the discrete systems' stiffness. Firstly and most naturally, we generalize SoftFEM by adding a least-square term to the mass bilinear form. Superconvergent results of rates <span><math><msup><mrow><mi>h</mi></mrow><mrow><mn>6</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>h</mi></mrow><mrow><mn>8</mn></mrow></msup></math></span> for eigenvalues are established for linear uniform elements; <span><math><msup><mrow><mi>h</mi></mrow><mrow><mn>8</mn></mrow></msup></math></span> is the highest order of convergence known in the literature. Secondly, we generalize SoftFEM by applying the blended Gaussian-type quadratures. We demonstrate further reductions in stiffness compared to traditional FEM and SoftFEM. The coercivity and analysis of the optimal error convergences follow the work of SoftFEM. Thus, this paper focuses on the numerical study of these generalizations. For linear and uniform elements, analytical eigenpairs, exact eigenvalue errors, and superconvergent error analysis are established. Various numerical examples demonstrate the potential of generalized SoftFEMs for spectral approximation, particularly in high-frequency regimes.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"184 ","pages":"Pages 168-184"},"PeriodicalIF":2.9,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143479161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-24DOI: 10.1016/j.camwa.2025.02.015
Xiran Cao , Zhengze Rong , Ping Lin , Liancun Zheng , Xuelan Zhang
The laser-induced thermal therapy (LITT) scheme has proved great efficacy in tumor treatment. Therefore, the research between the heat conduction problems of LITT has become a hot topic in recent years. To seek rational constitutive relations of heat flux and temperature which can describe the heat transfer behavior of LITT, we develop a novel distributed-order time fractional derivative model based on the dual-phase-lag (DPL) model and Pennes bio-heat conduction model in this paper. Physical parameters of the governing equation are approximated using experimental data. Formulated model considers a spectrum of memory and nonlocal characteristics based on the DPL model. Distributed-order integrals are approximated by the summation of multi-fractional terms and fractional derivatives are discretized by the L1 scheme. Source item is introduced into the governing equation to verify the correctness of the numerical methods. The influences of the physical parameters on the tissue temperature are discussed and analyzed in details. Results demonstrate that the proposed model truly performs better compared to the classical Fourier's law and DPL model in describing the heat conduction behavior of LITT.
{"title":"A novel distributed-order time fractional derivative model of laser-induced thermal therapy for deep-lying tumor","authors":"Xiran Cao , Zhengze Rong , Ping Lin , Liancun Zheng , Xuelan Zhang","doi":"10.1016/j.camwa.2025.02.015","DOIUrl":"10.1016/j.camwa.2025.02.015","url":null,"abstract":"<div><div>The laser-induced thermal therapy (LITT) scheme has proved great efficacy in tumor treatment. Therefore, the research between the heat conduction problems of LITT has become a hot topic in recent years. To seek rational constitutive relations of heat flux and temperature which can describe the heat transfer behavior of LITT, we develop a novel distributed-order time fractional derivative model based on the dual-phase-lag (DPL) model and Pennes bio-heat conduction model in this paper. Physical parameters of the governing equation are approximated using experimental data. Formulated model considers a spectrum of memory and nonlocal characteristics based on the DPL model. Distributed-order integrals are approximated by the summation of multi-fractional terms and fractional derivatives are discretized by the L1 scheme. Source item is introduced into the governing equation to verify the correctness of the numerical methods. The influences of the physical parameters on the tissue temperature are discussed and analyzed in details. Results demonstrate that the proposed model truly performs better compared to the classical Fourier's law and DPL model in describing the heat conduction behavior of LITT.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"184 ","pages":"Pages 107-115"},"PeriodicalIF":2.9,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143479132","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 : 2025-02-24DOI: 10.1016/j.camwa.2025.02.018
Xinyan Li
In this paper, we develop a numerical method for determining the potential in one and two dimensional fractional Calderón problems with a single measurement. Finite difference scheme is employed to discretize the fractional Laplacian, and the parameter reconstruction is formulated into a variational problem based on Tikhonov regularization to obtain a stable and accurate solution. Conjugate gradient method is utilized to solve the variational problem. Moreover, we also provide a suggestion to choose the regularization parameter. Numerical experiments are performed to illustrate the efficiency and effectiveness of the developed method and verify the theoretical results.
{"title":"A numerical method for reconstructing the potential in fractional Calderón problem with a single measurement","authors":"Xinyan Li","doi":"10.1016/j.camwa.2025.02.018","DOIUrl":"10.1016/j.camwa.2025.02.018","url":null,"abstract":"<div><div>In this paper, we develop a numerical method for determining the potential in one and two dimensional fractional Calderón problems with a single measurement. Finite difference scheme is employed to discretize the fractional Laplacian, and the parameter reconstruction is formulated into a variational problem based on Tikhonov regularization to obtain a stable and accurate solution. Conjugate gradient method is utilized to solve the variational problem. Moreover, we also provide a suggestion to choose the regularization parameter. Numerical experiments are performed to illustrate the efficiency and effectiveness of the developed method and verify the theoretical results.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"183 ","pages":"Pages 256-270"},"PeriodicalIF":2.9,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143478704","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 : 2025-02-21DOI: 10.1016/j.camwa.2025.02.009
Dingwen Deng, Yuxin Liang
<div><div>In this study, using implicit Euler method and second-order centered difference methods to approximate the first-order temporal and second-order spatial derivatives, respectively, introducing a stabilized term and applying <span><math><mi>u</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>−</mo><msup><mrow><mo>[</mo><mi>u</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>]</mo></mrow><mrow><mi>p</mi></mrow></msup><mi>u</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span> to approximate the nonlinear term <span><math><mi>u</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>−</mo><msup><mrow><mo>[</mo><mi>u</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>]</mo></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> at <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span>, a class of stabilized, non-negativity- and boundedness-preserving finite difference methods (FDMs) are derived for Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) equation. Here, <span><math><mi>u</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> denotes the exact solution of the original problem at <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>. In comparison with the existent maximum-principle-satisfying FDMs, our methods can further preserve the energy-dissipation property of the continuous problem with <span><math><mi>p</mi><mo>=</mo><mn>1</mn></math></span> or <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span>, and homogeneous Dirich
{"title":"Analysis of a class of stabilized and structure-preserving finite difference methods for Fisher-Kolmogorov-Petrovsky-Piscounov equation","authors":"Dingwen Deng, Yuxin Liang","doi":"10.1016/j.camwa.2025.02.009","DOIUrl":"10.1016/j.camwa.2025.02.009","url":null,"abstract":"<div><div>In this study, using implicit Euler method and second-order centered difference methods to approximate the first-order temporal and second-order spatial derivatives, respectively, introducing a stabilized term and applying <span><math><mi>u</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>−</mo><msup><mrow><mo>[</mo><mi>u</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>]</mo></mrow><mrow><mi>p</mi></mrow></msup><mi>u</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span> to approximate the nonlinear term <span><math><mi>u</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>−</mo><msup><mrow><mo>[</mo><mi>u</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>]</mo></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> at <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span>, a class of stabilized, non-negativity- and boundedness-preserving finite difference methods (FDMs) are derived for Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) equation. Here, <span><math><mi>u</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> denotes the exact solution of the original problem at <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>. In comparison with the existent maximum-principle-satisfying FDMs, our methods can further preserve the energy-dissipation property of the continuous problem with <span><math><mi>p</mi><mo>=</mo><mn>1</mn></math></span> or <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span>, and homogeneous Dirich","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"184 ","pages":"Pages 86-106"},"PeriodicalIF":2.9,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143465175","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 paper presents a novel physics-informed neural network (PINN) architecture that employs exponential basis functions (EBFs) to solve many boundary value problems. The EBFs are organized so that the PINN architecture may employ their simple differentiation features to solve partial differential equations (PDEs). The proposed approach has been meticulously investigated and compared to the conventional PINN in the solution of many instances involving Laplace and Helmholtz differential equations, as well as linear and nonlinear elasto-static problems. By utilizing EBFs, fewer interpolation functions are needed and the derivation rule can be explicitly applied, significantly reducing computation time. In addition, in some problems, PINN embedded with EBFs shows better precision near the boundaries of the problem, which is one of the disadvantages of conventional PINNs. Two specific applications of the presented method have been investigated, namely the inverse identification of material properties and the solution of complex-valued partial differential equations. The proposed method was found to be accurate and converging when applied to inverse problems. As shown, it is also suitable for the solution of PDEs containing both imaginary and real parts.
{"title":"On the use of exponential basis functions in the PINN architecture: An enhanced solution approach for the Laplace, Helmholtz, and elasto-static equations","authors":"Sadegh Ghamsari Esfahani, Bashir Movahedian, Saeid Sarrami, Mojtaba Azhari","doi":"10.1016/j.camwa.2025.02.004","DOIUrl":"10.1016/j.camwa.2025.02.004","url":null,"abstract":"<div><div>This paper presents a novel physics-informed neural network (PINN) architecture that employs exponential basis functions (EBFs) to solve many boundary value problems. The EBFs are organized so that the PINN architecture may employ their simple differentiation features to solve partial differential equations (PDEs). The proposed approach has been meticulously investigated and compared to the conventional PINN in the solution of many instances involving Laplace and Helmholtz differential equations, as well as linear and nonlinear elasto-static problems. By utilizing EBFs, fewer interpolation functions are needed and the derivation rule can be explicitly applied, significantly reducing computation time. In addition, in some problems, PINN embedded with EBFs shows better precision near the boundaries of the problem, which is one of the disadvantages of conventional PINNs. Two specific applications of the presented method have been investigated, namely the inverse identification of material properties and the solution of complex-valued partial differential equations. The proposed method was found to be accurate and converging when applied to inverse problems. As shown, it is also suitable for the solution of PDEs containing both imaginary and real parts.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"183 ","pages":"Pages 234-255"},"PeriodicalIF":2.9,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445045","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 : 2025-02-20DOI: 10.1016/j.camwa.2025.02.010
María Gabriela Armentano , Claudio Padra , Mario Scheble
In this paper we introduce and analyze an hp finite element method to solve a non-standard spectral problem in a curved plane domain using curved elements. This problem arises from nuclear engineering: the vibration of elastically mounted tubes immersed in a cavity filled with fluid. The eigenvalue problem is presented in a proper setting and we prove, under appropriate assumptions about the curved domain, the convergence of the method and a priori error estimates for the eigenfunctions and the eigenvalues. We define an efficiency and reliability a posteriori error indicator of the residual type up to higher order terms. We analyze in detail the symmetric case, and we propose and efficient approach which allows us simplify the eigenvalue problem and solve efficiently the case of multiples eigenvalues. Finally, we present an hp adaptive algorithm and some numerical tests which show the performance of the scheme, including evidence of exponential convergence.
{"title":"An hp finite element a posteriori analysis of a non-standard eigenvalue problem arising in fluid-solid vibrations on curved domains","authors":"María Gabriela Armentano , Claudio Padra , Mario Scheble","doi":"10.1016/j.camwa.2025.02.010","DOIUrl":"10.1016/j.camwa.2025.02.010","url":null,"abstract":"<div><div>In this paper we introduce and analyze an <em>hp</em> finite element method to solve a non-standard spectral problem in a curved plane domain using curved elements. This problem arises from nuclear engineering: the vibration of elastically mounted tubes immersed in a cavity filled with fluid. The eigenvalue problem is presented in a proper setting and we prove, under appropriate assumptions about the curved domain, the convergence of the method and a priori error estimates for the eigenfunctions and the eigenvalues. We define an efficiency and reliability a posteriori error indicator of the residual type up to higher order terms. We analyze in detail the symmetric case, and we propose and efficient approach which allows us simplify the eigenvalue problem and solve efficiently the case of multiples eigenvalues. Finally, we present an <em>hp</em> adaptive algorithm and some numerical tests which show the performance of the scheme, including evidence of exponential convergence.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"184 ","pages":"Pages 65-85"},"PeriodicalIF":2.9,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455117","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 : 2025-02-19DOI: 10.1016/j.camwa.2025.02.011
Yuxiang Liang, Shun Zhang
In this paper, for the generalized Darcy problem (an elliptic equation with discontinuous coefficients), we study a special partial Least-Squares (Galerkin-least-squares) method, known as the augmented mixed finite element method, and its relationship to the standard least-squares finite element method (LSFEM). Two versions of augmented mixed finite element methods are proposed in the paper with robust a priori and a posteriori error estimates. Augmented mixed finite element methods and the standard LSFEM uses the same a posteriori error estimator: the evaluations of numerical solutions at the corresponding least-squares functionals. As partial least-squares methods, the augmented mixed finite element methods are more flexible than the original LSFEMs. As comparisons, we discuss the mild non-robustness of a priori and a posteriori error estimates of the original LSFEMs. A special case that the -based LSFEM is robust is also presented for the first time. Extensive numerical experiments are presented to verify our findings.
{"title":"Least-squares versus partial least-squares finite element methods: Robust a priori and a posteriori error estimates of augmented mixed finite element methods","authors":"Yuxiang Liang, Shun Zhang","doi":"10.1016/j.camwa.2025.02.011","DOIUrl":"10.1016/j.camwa.2025.02.011","url":null,"abstract":"<div><div>In this paper, for the generalized Darcy problem (an elliptic equation with discontinuous coefficients), we study a special partial Least-Squares (Galerkin-least-squares) method, known as the augmented mixed finite element method, and its relationship to the standard least-squares finite element method (LSFEM). Two versions of augmented mixed finite element methods are proposed in the paper with robust a priori and a posteriori error estimates. Augmented mixed finite element methods and the standard LSFEM uses the same a posteriori error estimator: the evaluations of numerical solutions at the corresponding least-squares functionals. As partial least-squares methods, the augmented mixed finite element methods are more flexible than the original LSFEMs. As comparisons, we discuss the mild non-robustness of a priori and a posteriori error estimates of the original LSFEMs. A special case that the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-based LSFEM is robust is also presented for the first time. Extensive numerical experiments are presented to verify our findings.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"184 ","pages":"Pages 45-64"},"PeriodicalIF":2.9,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143444522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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