Pub Date : 2024-11-27DOI: 10.1016/j.camwa.2024.11.022
János Karátson , Stanislav Sysala , Michal Béreš
This paper is devoted to the extension of a quasi-Newton/variable preconditioning (QNVP) method to non-smooth problems, motivated by elasto-plastic models. Two approaches are discussed: the first one is carried out via regularized approximations of the nonsmooth problem, and the second one gives an extension to nonsmooth operators in order to be applied directly. Convergence analysis is presented for both variants. Then these abstract methods are applied to elasto-plasticity where two different variants of QNVP are investigated and combined with the deflated conjugate gradient and aggregation-based algebraic multigrid methods. The convergence results are illustrated on numerical examples in 3D inspired by real-life problems, and they demonstrate that the suggested QNVP methods are competitive with the standard Newton method. Well-documented Matlab codes on elasto-plasticity are used and enriched by the suggested methods.
{"title":"Quasi-Newton iterative solution approaches for nonsmooth elliptic operators with applications to elasto-plasticity","authors":"János Karátson , Stanislav Sysala , Michal Béreš","doi":"10.1016/j.camwa.2024.11.022","DOIUrl":"10.1016/j.camwa.2024.11.022","url":null,"abstract":"<div><div>This paper is devoted to the extension of a quasi-Newton/variable preconditioning (QNVP) method to non-smooth problems, motivated by elasto-plastic models. Two approaches are discussed: the first one is carried out via regularized approximations of the nonsmooth problem, and the second one gives an extension to nonsmooth operators in order to be applied directly. Convergence analysis is presented for both variants. Then these abstract methods are applied to elasto-plasticity where two different variants of QNVP are investigated and combined with the deflated conjugate gradient and aggregation-based algebraic multigrid methods. The convergence results are illustrated on numerical examples in 3D inspired by real-life problems, and they demonstrate that the suggested QNVP methods are competitive with the standard Newton method. Well-documented Matlab codes on elasto-plasticity are used and enriched by the suggested methods.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"178 ","pages":"Pages 61-80"},"PeriodicalIF":2.9,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142721493","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 : 2024-11-26DOI: 10.1016/j.camwa.2024.11.024
Jun Liu , Chenxi Ji , Wenbin Ye , Lei Gan , Lei Qin , Quansheng Zang , Haibo Wang
In this paper, a modified scaled boundary finite element method (SBFEM) is developed to study static and vibration behaviors of laminated conical shells under the conical coordinate system. In the modified SBFEM, the geometry of the conical shell is defined entirely by scaling the internal surface of the structure. This approach eliminates geometric errors caused by discretization, thereby enhancing modeling accuracy. The three-dimensional problem is simplified to a two-dimensional analysis since discretization is only applied to the boundary of the computational domain. Additionally, the semi-analytic property of the SBFEM allows for the derivation of a linear analytical solution for the laminated conical shell in the radial direction. First, a scaled boundary coordinate system for the scaling surface is established, and a second-order scaled boundary finite element governing equation with variable coefficients is derived for a single layer of the conical shell using the principle of virtual work. Next, the governing equation is transformed into a first-order system by introducing a combined vector of displacement and nodal force, and the stiffness matrices for each layer of the laminated conical shell are obtained using the precise integration method. Finally, an overall analysis of the laminated structure is conducted by assembling each single-layer structure while considering the continuity boundary condition at interfaces. Static and vibration analyses of laminated conical shells are conducted, and the results are compared with those from the literature to demonstrate the adaptability and convergence of the proposed method. Several numerical examples are presented to examine the effects of various geometric parameters, such as thickness, length, semi-vertex angles, layup directions, and stacking sequences, on the responses of the structure.
{"title":"Static and vibration analyses of laminated conical shells under various boundary conditions using a modified scaled boundary finite element method","authors":"Jun Liu , Chenxi Ji , Wenbin Ye , Lei Gan , Lei Qin , Quansheng Zang , Haibo Wang","doi":"10.1016/j.camwa.2024.11.024","DOIUrl":"10.1016/j.camwa.2024.11.024","url":null,"abstract":"<div><div>In this paper, a modified scaled boundary finite element method (SBFEM) is developed to study static and vibration behaviors of laminated conical shells under the conical coordinate system. In the modified SBFEM, the geometry of the conical shell is defined entirely by scaling the internal surface of the structure. This approach eliminates geometric errors caused by discretization, thereby enhancing modeling accuracy. The three-dimensional problem is simplified to a two-dimensional analysis since discretization is only applied to the boundary of the computational domain. Additionally, the semi-analytic property of the SBFEM allows for the derivation of a linear analytical solution for the laminated conical shell in the radial direction. First, a scaled boundary coordinate system for the scaling surface is established, and a second-order scaled boundary finite element governing equation with variable coefficients is derived for a single layer of the conical shell using the principle of virtual work. Next, the governing equation is transformed into a first-order system by introducing a combined vector of displacement and nodal force, and the stiffness matrices for each layer of the laminated conical shell are obtained using the precise integration method. Finally, an overall analysis of the laminated structure is conducted by assembling each single-layer structure while considering the continuity boundary condition at interfaces. Static and vibration analyses of laminated conical shells are conducted, and the results are compared with those from the literature to demonstrate the adaptability and convergence of the proposed method. Several numerical examples are presented to examine the effects of various geometric parameters, such as thickness, length, semi-vertex angles, layup directions, and stacking sequences, on the responses of the structure.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"177 ","pages":"Pages 147-166"},"PeriodicalIF":2.9,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698885","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 : 2024-11-26DOI: 10.1016/j.camwa.2024.11.014
Gang Peng , Yuan Li
In this paper, an energy stable bound-preserving finite volume scheme is constructed for the Allen-Cahn equation. The first-order operator splitting method is used to split the original equation into a nonlinear equation and a heat equation in each time interval. The nonlinear equation is solved by the explicit scheme, and the heat equation is discretized by the extremum-preserving scheme. The harmonic averaging points on cell facets are employed to define auxiliary unknowns, which enable our discrete scheme to be applicable to unstructured meshes. The energy stable and bound-preserving analysis of the finite volume scheme are also presented. Numerical experiments illustrate that this linear numerical scheme is practical and accurate in solving the Allen-Cahn equation.
{"title":"An energy stable bound-preserving finite volume scheme for the Allen-Cahn equation based on operator splitting method","authors":"Gang Peng , Yuan Li","doi":"10.1016/j.camwa.2024.11.014","DOIUrl":"10.1016/j.camwa.2024.11.014","url":null,"abstract":"<div><div>In this paper, an energy stable bound-preserving finite volume scheme is constructed for the Allen-Cahn equation. The first-order operator splitting method is used to split the original equation into a nonlinear equation and a heat equation in each time interval. The nonlinear equation is solved by the explicit scheme, and the heat equation is discretized by the extremum-preserving scheme. The harmonic averaging points on cell facets are employed to define auxiliary unknowns, which enable our discrete scheme to be applicable to unstructured meshes. The energy stable and bound-preserving analysis of the finite volume scheme are also presented. Numerical experiments illustrate that this linear numerical scheme is practical and accurate in solving the Allen-Cahn equation.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"178 ","pages":"Pages 47-60"},"PeriodicalIF":2.9,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142718618","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 : 2024-11-25DOI: 10.1016/j.camwa.2024.11.023
Lu Wang , Minfu Feng
This paper presents a simplified weak Galerkin (SWG) method for solving the elastodynamic problem and its reduced-order model (ROM) using the proper orthogonal decomposition (POD) technique. The SWG method allows for the use of polygonal meshes. It only utilizes degrees of freedom associated with the boundary, reducing computational complexity compared to the classical weak Galerkin method. Moreover, we apply the POD technique to develop a POD-SWG-ROM for the problem, further enhancing the computational efficiency. Then, to discretize in time, we utilize a θ-scheme, where the scheme is explicit when and implicit when . We establish the theoretical analysis of the semi-discrete scheme and the fully-discrete θ scheme. The theoretical analysis demonstrates that the method is locking-free, and the convergence rate in the and norms is and respectively. Finally, we verify the theoretical analysis through numerical tests and effectively simulate the propagation of elastic waves under polygonal meshes. Moreover, the proposed POD-SWG-ROM can significantly improve computational efficiency.
{"title":"The simplified weak Galerkin method with θ scheme and its reduced-order model for the elastodynamic problem on polygonal mesh","authors":"Lu Wang , Minfu Feng","doi":"10.1016/j.camwa.2024.11.023","DOIUrl":"10.1016/j.camwa.2024.11.023","url":null,"abstract":"<div><div>This paper presents a simplified weak Galerkin (SWG) method for solving the elastodynamic problem and its reduced-order model (ROM) using the proper orthogonal decomposition (POD) technique. The SWG method allows for the use of polygonal meshes. It only utilizes degrees of freedom associated with the boundary, reducing computational complexity compared to the classical weak Galerkin method. Moreover, we apply the POD technique to develop a POD-SWG-ROM for the problem, further enhancing the computational efficiency. Then, to discretize in time, we utilize a <em>θ</em>-scheme, where the scheme is explicit when <span><math><mn>0</mn><mo>≤</mo><mi>θ</mi><mo><</mo><mn>1</mn><mo>/</mo><mn>4</mn></math></span> and implicit when <span><math><mn>1</mn><mo>/</mo><mn>4</mn><mo>≤</mo><mi>θ</mi><mo>≤</mo><mn>1</mn><mo>/</mo><mn>2</mn></math></span>. We establish the theoretical analysis of the semi-discrete scheme and the fully-discrete <em>θ</em> scheme. The theoretical analysis demonstrates that the method is locking-free, and the convergence rate in the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> norms is <span><math><mi>O</mi><mo>(</mo><mi>Δ</mi><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></math></span> and <span><math><mi>O</mi><mo>(</mo><mi>Δ</mi><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> respectively. Finally, we verify the theoretical analysis through numerical tests and effectively simulate the propagation of elastic waves under polygonal meshes. Moreover, the proposed POD-SWG-ROM can significantly improve computational efficiency.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"178 ","pages":"Pages 19-46"},"PeriodicalIF":2.9,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699957","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 : 2024-11-23DOI: 10.1016/j.camwa.2024.11.020
Wensi Wang , Hailing Xuan , Xiaoliang Cheng , Kewei Liang
We present a mathematical model describing the equilibrium of a viscoelastic body with long-term memory in frictional contact with a sliding foundation. The process is quasistatic, and material damage resulting from excessive stress or strain is captured by a damage function. We assume the material is inhomogeneous, leading to multiple contact boundary conditions. The contact interface is partitioned into two segments: One part takes into account the wear of the contact surface, utilizing Archard's law. Here, contact is modeled with a normal compliance condition with unilateral constraints, coupled with a sliding version of Coulomb's law of dry friction. In the other part, contact is modeled with a nonmonotone condition involving normal compliance and a subdifferential frictional boundary condition. Variational formulation of the model is governed by a coupled system consisting of a variational–hemivariational inequality for the displacement field, a parabolic variational inequality for the damage field and an integral equation for the wear function. We study a fully discrete scheme for numerical approximation with an error estimation of the solution to this problem. Optimal error estimates for the linear finite element method are derived, followed by numerical simulations illustrating the behavior of the model.
{"title":"Numerical analysis and simulation of a quasistatic frictional bilateral contact problem with damage, long-term memory and wear","authors":"Wensi Wang , Hailing Xuan , Xiaoliang Cheng , Kewei Liang","doi":"10.1016/j.camwa.2024.11.020","DOIUrl":"10.1016/j.camwa.2024.11.020","url":null,"abstract":"<div><div>We present a mathematical model describing the equilibrium of a viscoelastic body with long-term memory in frictional contact with a sliding foundation. The process is quasistatic, and material damage resulting from excessive stress or strain is captured by a damage function. We assume the material is inhomogeneous, leading to multiple contact boundary conditions. The contact interface is partitioned into two segments: One part takes into account the wear of the contact surface, utilizing Archard's law. Here, contact is modeled with a normal compliance condition with unilateral constraints, coupled with a sliding version of Coulomb's law of dry friction. In the other part, contact is modeled with a nonmonotone condition involving normal compliance and a subdifferential frictional boundary condition. Variational formulation of the model is governed by a coupled system consisting of a variational–hemivariational inequality for the displacement field, a parabolic variational inequality for the damage field and an integral equation for the wear function. We study a fully discrete scheme for numerical approximation with an error estimation of the solution to this problem. Optimal error estimates for the linear finite element method are derived, followed by numerical simulations illustrating the behavior of the model.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"177 ","pages":"Pages 130-146"},"PeriodicalIF":2.9,"publicationDate":"2024-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698884","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 : 2024-11-22DOI: 10.1016/j.camwa.2024.11.019
Antti Autio, Antti Hannukainen
In this work, a subspace method is proposed for efficient solution of parametric linear systems with a symmetric and positive definite coefficient matrix of the form . The motivation is to use the method for solution of linear systems appearing when solving parameter dependent elliptic PDEs using the finite element method (FEM). In the proposed method, one first computes a method subspace and then uses it to approximately solve the linear system for any parameter vector. The method subspace is designed in such a way that it contains the -term truncated Neumann series approximation of the solution to desired accuracy for any admissible parameter vector. This allows us to use the best approximation property of subspace methods to show that the subspace solution is at least as accurate as the truncated Neumann series approximation. The performance of the method is demonstrated by numerical examples with the parametric diffusion equation. In these examples, the method yields much smaller errors than anticipated by the Neumann series based error analysis. We study this phenomenon in some special cases.
{"title":"A subspace method based on the Neumann series for the solution of parametric linear systems","authors":"Antti Autio, Antti Hannukainen","doi":"10.1016/j.camwa.2024.11.019","DOIUrl":"10.1016/j.camwa.2024.11.019","url":null,"abstract":"<div><div>In this work, a subspace method is proposed for efficient solution of parametric linear systems with a symmetric and positive definite coefficient matrix of the form <span><math><mi>I</mi><mo>−</mo><mi>A</mi><mo>(</mo><mi>σ</mi><mo>)</mo></math></span>. The motivation is to use the method for solution of linear systems appearing when solving parameter dependent elliptic PDEs using the finite element method (FEM). In the proposed method, one first computes a method subspace and then uses it to approximately solve the linear system for any parameter vector. The method subspace is designed in such a way that it contains the <span><math><mi>j</mi><mo>+</mo><mn>1</mn></math></span>-term truncated Neumann series approximation of the solution to desired accuracy for any admissible parameter vector. This allows us to use the best approximation property of subspace methods to show that the subspace solution is at least as accurate as the truncated Neumann series approximation. The performance of the method is demonstrated by numerical examples with the parametric diffusion equation. In these examples, the method yields much smaller errors than anticipated by the Neumann series based error analysis. We study this phenomenon in some special cases.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"178 ","pages":"Pages 1-18"},"PeriodicalIF":2.9,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699956","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 : 2024-11-20DOI: 10.1016/j.camwa.2024.11.015
Ihor Borachok , Roman Chapko , Leonidas Mindrinos
In this study, we propose two different approaches of a two-step method for solving a system combining the heat and the wave equations. Our focus centers on the transmission problem in two dimensions, with a primary objective of numerically characterizing the distribution of temperature and pressure. First we apply a semi-discretization with respect to time by using the Laguerre transformation. This leads to a sequence of elliptic problems that are fully discretized either by the boundary integral equation method or by the fundamental sequences method. The presented numerical examples justify the applicability and efficiency of both schemes.
{"title":"Two-step numerical methods for a coupled parabolic-hyperbolic transmission problem","authors":"Ihor Borachok , Roman Chapko , Leonidas Mindrinos","doi":"10.1016/j.camwa.2024.11.015","DOIUrl":"10.1016/j.camwa.2024.11.015","url":null,"abstract":"<div><div>In this study, we propose two different approaches of a two-step method for solving a system combining the heat and the wave equations. Our focus centers on the transmission problem in two dimensions, with a primary objective of numerically characterizing the distribution of temperature and pressure. First we apply a semi-discretization with respect to time by using the Laguerre transformation. This leads to a sequence of elliptic problems that are fully discretized either by the boundary integral equation method or by the fundamental sequences method. The presented numerical examples justify the applicability and efficiency of both schemes.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"177 ","pages":"Pages 115-129"},"PeriodicalIF":2.9,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698883","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 : 2024-11-19DOI: 10.1016/j.camwa.2024.11.017
Stefano Berrone , Andrea Borio , Francesca Marcon
We analyze the first order Enlarged Enhancement Virtual Element Method (E2VEM) for the Poisson problem. The method allows the definition of bilinear forms that do not require a stabilization term, thanks to the exploitation of higher order polynomial projections that are made computable by suitably enlarging the enhancement property (from which comes the prefix E2) of local virtual spaces. We provide a sufficient condition for the well-posedness and optimal order a priori error estimates. We present numerical tests on convex and non-convex polygonal meshes that confirm the robustness of the method and the theoretical convergence rates.
{"title":"Lowest order stabilization free virtual element method for the 2D Poisson equation","authors":"Stefano Berrone , Andrea Borio , Francesca Marcon","doi":"10.1016/j.camwa.2024.11.017","DOIUrl":"10.1016/j.camwa.2024.11.017","url":null,"abstract":"<div><div>We analyze the first order Enlarged Enhancement Virtual Element Method (E<sup>2</sup>VEM) for the Poisson problem. The method allows the definition of bilinear forms that do not require a stabilization term, thanks to the exploitation of higher order polynomial projections that are made computable by suitably enlarging the enhancement property (from which comes the prefix E<sup>2</sup>) of local virtual spaces. We provide a sufficient condition for the well-posedness and optimal order a priori error estimates. We present numerical tests on convex and non-convex polygonal meshes that confirm the robustness of the method and the theoretical convergence rates.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"177 ","pages":"Pages 78-99"},"PeriodicalIF":2.9,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142673449","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 : 2024-11-19DOI: 10.1016/j.camwa.2024.11.013
Yajun Liu, Yuanyang Qiao, Xinlong Feng
A semi-Lagrangian radial basis function partition of unity (RBF-PU) closest point method is designed for solving advection-diffusion equations on surfaces. This new meshfree method combines the semi-Lagrangian method with the RBF-PU closest point method. The semi-Lagrangian RBF-PU closest point method traces the departure point backwards along the velocity field based on patches at each time step. Therefore, it saves the computational cost compared with the semi-Lagrangian radial basis function finite difference (RBF-FD) closest point method. Our proposed RBF-PU closest point method for approximating the Laplace-Beltrami operator has two main advantages over the RBF-FD closest point method. Firstly, the RBF-PU closest point method to construct the local influence domain is not required to identify the size of the computational tube. Therefore, our method is more concise and easier to implement. Secondly, the RBF-PU closest point method not only improves the accuracy but also saves computational costs, and we confirm the advantage by testing differential accuracy. Numerical experiments verify the convergence and effectiveness of the semi-Lagrangian RBF-PU closest point method.
{"title":"A semi-Lagrangian radial basis function partition of unity closest point method for advection-diffusion equations on surfaces","authors":"Yajun Liu, Yuanyang Qiao, Xinlong Feng","doi":"10.1016/j.camwa.2024.11.013","DOIUrl":"10.1016/j.camwa.2024.11.013","url":null,"abstract":"<div><div>A semi-Lagrangian radial basis function partition of unity (RBF-PU) closest point method is designed for solving advection-diffusion equations on surfaces. This new meshfree method combines the semi-Lagrangian method with the RBF-PU closest point method. The semi-Lagrangian RBF-PU closest point method traces the departure point backwards along the velocity field based on patches at each time step. Therefore, it saves the computational cost compared with the semi-Lagrangian radial basis function finite difference (RBF-FD) closest point method. Our proposed RBF-PU closest point method for approximating the Laplace-Beltrami operator has two main advantages over the RBF-FD closest point method. Firstly, the RBF-PU closest point method to construct the local influence domain is not required to identify the size of the computational tube. Therefore, our method is more concise and easier to implement. Secondly, the RBF-PU closest point method not only improves the accuracy but also saves computational costs, and we confirm the advantage by testing differential accuracy. Numerical experiments verify the convergence and effectiveness of the semi-Lagrangian RBF-PU closest point method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"177 ","pages":"Pages 100-114"},"PeriodicalIF":2.9,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142673451","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 : 2024-11-19DOI: 10.1016/j.camwa.2024.11.016
Xiaojuan Liu , Maojun Li , Kun Wang , Jiangming Xie
As one of the most popular artificial boundary conditions, the Dirichlet-to-Neumann (DtN) boundary condition has been widely developed and investigated for solving the exterior wave scattering problems. This work studies the application of a Fourier series DtN map for the elastic scattering problem. The infinite series of the DtN map requires to be truncated in the practical numerical application, and then the well-posedness of the resulting boundary value problem (BVP) becomes a challenging issue. By introducing a corresponding eigensystem to the bilinear form together with appropriate truncated norm estimates, we prove the well-posedness of the corresponding BVP in a weak sense. In addition, a priori error estimates that incorporate the effects of the finite element discretization and the truncation of infinite series are derived. Finally, numerical tests are implemented to validate the theoretical results.
{"title":"Well-posedness and finite element analysis for the elastic scattering problem with a modified DtN map","authors":"Xiaojuan Liu , Maojun Li , Kun Wang , Jiangming Xie","doi":"10.1016/j.camwa.2024.11.016","DOIUrl":"10.1016/j.camwa.2024.11.016","url":null,"abstract":"<div><div>As one of the most popular artificial boundary conditions, the Dirichlet-to-Neumann (DtN) boundary condition has been widely developed and investigated for solving the exterior wave scattering problems. This work studies the application of a Fourier series DtN map for the elastic scattering problem. The infinite series of the DtN map requires to be truncated in the practical numerical application, and then the well-posedness of the resulting boundary value problem (BVP) becomes a challenging issue. By introducing a corresponding eigensystem to the bilinear form together with appropriate truncated norm estimates, we prove the well-posedness of the corresponding BVP in a weak sense. In addition, a priori error estimates that incorporate the effects of the finite element discretization and the truncation of infinite series are derived. Finally, numerical tests are implemented to validate the theoretical results.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"177 ","pages":"Pages 58-77"},"PeriodicalIF":2.9,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142673450","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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