Pub Date : 2024-11-10DOI: 10.1016/j.cam.2024.116371
Burcu Fedakar , Ilhame Amirali , Muhammet Enes Durmaz , Gabil M. Amiraliyev
This work deals with the initial-value problem for a second-order neutral Volterra integro-differential equation. First, we give the stability inequality indicating stability of the problem with respect to the right-side and initial conditions. Further, we develop a finite difference method that uses for differential part second difference derivative, for the integral part appropriate composite trapezoidal and midpoint rectangle rules followed by second-order accurate difference quantities at intermediate points. Error estimate for the approximate solution is established. In support of theoretical results, numerical results are performed by employing the proposed numerical technique.
{"title":"Numerical solutions for second-order neutral volterra integro-differential equations: Stability analysis and finite difference method","authors":"Burcu Fedakar , Ilhame Amirali , Muhammet Enes Durmaz , Gabil M. Amiraliyev","doi":"10.1016/j.cam.2024.116371","DOIUrl":"10.1016/j.cam.2024.116371","url":null,"abstract":"<div><div>This work deals with the initial-value problem for a second-order neutral Volterra integro-differential equation. First, we give the stability inequality indicating stability of the problem with respect to the right-side and initial conditions. Further, we develop a finite difference method that uses for differential part second difference derivative, for the integral part appropriate composite trapezoidal and midpoint rectangle rules followed by second-order accurate difference quantities at intermediate points. Error estimate for the approximate solution is established. In support of theoretical results, numerical results are performed by employing the proposed numerical technique.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"459 ","pages":"Article 116371"},"PeriodicalIF":2.1,"publicationDate":"2024-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660936","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 : 2024-11-10DOI: 10.1016/j.cam.2024.116366
Wei Xu , Weimin Han , Ting Li , Ziping Huang
A class of evolutionary variational–hemivariational inequalities with a convex constraint is studied in this paper. An inequality in this class involves a first-order derivative and a history-dependent operator. Existence and uniqueness of a solution to the inequality is established by the Rothe method, in which the first-order temporal derivative is approximated by backward Euler’s formula, and the history-dependent operator is approximated by a modified left endpoint rule. The proof of the result relies on basic results in functional analysis only, and it does not require the notion of pseudomonotone operators and abstract surjectivity results for such operators, used in other papers on the Rothe method for other evolutionary variational–hemivariational inequalities. Moreover, a Lipschitz continuous dependence conclusion of the solution on the right-hand side is proved. Finally, a new frictional contact problem for viscoelastic material is discussed, which illustrates an application of the theoretical results.
{"title":"Well-posedness of a class of evolutionary variational–hemivariational inequalities in contact mechanics","authors":"Wei Xu , Weimin Han , Ting Li , Ziping Huang","doi":"10.1016/j.cam.2024.116366","DOIUrl":"10.1016/j.cam.2024.116366","url":null,"abstract":"<div><div>A class of evolutionary variational–hemivariational inequalities with a convex constraint is studied in this paper. An inequality in this class involves a first-order derivative and a history-dependent operator. Existence and uniqueness of a solution to the inequality is established by the Rothe method, in which the first-order temporal derivative is approximated by backward Euler’s formula, and the history-dependent operator is approximated by a modified left endpoint rule. The proof of the result relies on basic results in functional analysis only, and it does not require the notion of pseudomonotone operators and abstract surjectivity results for such operators, used in other papers on the Rothe method for other evolutionary variational–hemivariational inequalities. Moreover, a Lipschitz continuous dependence conclusion of the solution on the right-hand side is proved. Finally, a new frictional contact problem for viscoelastic material is discussed, which illustrates an application of the theoretical results.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"459 ","pages":"Article 116366"},"PeriodicalIF":2.1,"publicationDate":"2024-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660933","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 : 2024-11-09DOI: 10.1016/j.cam.2024.116358
Sk. Safique Ahmad, Pinki Khatun
This paper proposes a new parameterized enhanced shift-splitting (PESS) preconditioner to solve the three-by-three block saddle point problem (SPP). Additionally, we introduce a local PESS (LPESS) preconditioner by relaxing the PESS preconditioner. Necessary and sufficient criteria are established for the convergence of the proposed PESS iterative process for any initial guess. Furthermore, we meticulously investigate the spectral bounds of the PESS and LPESS preconditioned matrices. Moreover, empirical investigations have been performed for the sensitivity analysis of the proposed PESS preconditioner, which unveils its robustness. Numerical experiments are carried out to demonstrate the enhanced efficiency and robustness of the proposed PESS and LPESS preconditioners compared to the existing state-of-the-art preconditioners.
{"title":"A robust parameterized enhanced shift-splitting preconditioner for three-by-three block saddle point problems","authors":"Sk. Safique Ahmad, Pinki Khatun","doi":"10.1016/j.cam.2024.116358","DOIUrl":"10.1016/j.cam.2024.116358","url":null,"abstract":"<div><div>This paper proposes a new parameterized enhanced shift-splitting <em>(PESS)</em> preconditioner to solve the three-by-three block saddle point problem (<em>SPP</em>). Additionally, we introduce a local <em>PESS</em> (<em>LPESS</em>) preconditioner by relaxing the <em>PESS</em> preconditioner. Necessary and sufficient criteria are established for the convergence of the proposed <em>PESS</em> iterative process for any initial guess. Furthermore, we meticulously investigate the spectral bounds of the <em>PESS</em> and <em>LPESS</em> preconditioned matrices. Moreover, empirical investigations have been performed for the sensitivity analysis of the proposed <em>PESS</em> preconditioner, which unveils its robustness. Numerical experiments are carried out to demonstrate the enhanced efficiency and robustness of the proposed <em>PESS</em> and <em>LPESS</em> preconditioners compared to the existing state-of-the-art preconditioners.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"459 ","pages":"Article 116358"},"PeriodicalIF":2.1,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661498","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 : 2024-11-09DOI: 10.1016/j.cam.2024.116370
Yanping Chen , Hanzhang Hu
A two-grid finite element method is developed for solving space-fractional nonlinear Schrödinger equations. The finite element solution in -norm is proved bounded without any time-step size conditions (dependent on spatial-step size). Then, the optimal order error estimations of the two-grid solution in the -norm are proved without any time-step size conditions. Finally, the theoretical results are verified by numerical experiments.
{"title":"Two-grid finite element methods for space-fractional nonlinear Schrödinger equations","authors":"Yanping Chen , Hanzhang Hu","doi":"10.1016/j.cam.2024.116370","DOIUrl":"10.1016/j.cam.2024.116370","url":null,"abstract":"<div><div>A two-grid finite element method is developed for solving space-fractional nonlinear Schrödinger equations. The finite element solution in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span>-norm is proved bounded without any time-step size conditions (dependent on spatial-step size). Then, the optimal order error estimations of the two-grid solution in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-norm are proved without any time-step size conditions. Finally, the theoretical results are verified by numerical experiments.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"459 ","pages":"Article 116370"},"PeriodicalIF":2.1,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661494","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 : 2024-11-09DOI: 10.1016/j.cam.2024.116374
Anna Piterskaya, Mikael Mortensen
The article discusses two spectral methods, namely the Galerkin and the Petrov–Galerkin methods, for linear stability analysis of magneto-hydrodynamic (MHD) equations describing the flow of an electrically conducting fluid in the presence of a tangential magnetic field. The stability and spectral accuracy of both methods have been compared by examining the most unstable eigensolution of the Orr–Sommerfeld (OS) and induction equations. The Petrov–Galerkin spectral method (PGSM) used in this work has been developed by choosing function spaces and basis functions that always lead to banded coefficient matrices. The Galerkin spectral method (GSM), on the contrary, leads to dense matrices when Chebyshev polynomials are utilized in a weighted inner product space. We have found that both the GSM and the PGSM can produce results with minimal round-off errors, as confirmed by computing the most unstable eigenvalue of the OS equations (Re ) to 14 decimal places of accuracy in double precision. We show that with properly scaled basis functions the GSM leads to coefficient matrices with bounded condition numbers, both for the OS equation and for the coupled OS and induction equations. This allows to achieve accurate results with double precision for any number of for both the GSM and the PGSM. The analysis of the different behavior of the condition numbers suggests that the proposed two methods, based on the Chebyshev polynomials, can become a useful computer-based tool that is capable of finding a numerical solution to both the hydrodynamic and the MHD equations at very high Reynolds numbers.
{"title":"A study of the Orr–Sommerfeld and induction equations by Galerkin and Petrov–Galerkin spectral methods utilizing Chebyshev polynomials","authors":"Anna Piterskaya, Mikael Mortensen","doi":"10.1016/j.cam.2024.116374","DOIUrl":"10.1016/j.cam.2024.116374","url":null,"abstract":"<div><div>The article discusses two spectral methods, namely the Galerkin and the Petrov–Galerkin methods, for linear stability analysis of magneto-hydrodynamic (MHD) equations describing the flow of an electrically conducting fluid in the presence of a tangential magnetic field. The stability and spectral accuracy of both methods have been compared by examining the most unstable eigensolution of the Orr–Sommerfeld (OS) and induction equations. The Petrov–Galerkin spectral method (PGSM) used in this work has been developed by choosing function spaces and basis functions that always lead to banded coefficient matrices. The Galerkin spectral method (GSM), on the contrary, leads to dense matrices when Chebyshev polynomials are utilized in a weighted inner product space. We have found that both the GSM and the PGSM can produce results with minimal round-off errors, as confirmed by computing the most unstable eigenvalue of the OS equations (Re <span><math><mrow><mo>=</mo><mn>1</mn><msup><mrow><mn>0</mn></mrow><mrow><mn>4</mn></mrow></msup></mrow></math></span>) to 14 decimal places of accuracy in double precision. We show that with properly scaled basis functions the GSM leads to coefficient matrices with bounded condition numbers, both for the OS equation and for the coupled OS and induction equations. This allows to achieve accurate results with double precision for any number of <span><math><mi>N</mi></math></span> for both the GSM and the PGSM. The analysis of the different behavior of the condition numbers suggests that the proposed two methods, based on the Chebyshev polynomials, can become a useful computer-based tool that is capable of finding a numerical solution to both the hydrodynamic and the MHD equations at very high Reynolds numbers.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"459 ","pages":"Article 116374"},"PeriodicalIF":2.1,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661492","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 : 2024-11-09DOI: 10.1016/j.cam.2024.116355
Dirk Martin , Gundolf Haase , Günter Offner
The inverse matrix of radial basis function (RBF) interpolation systems can be stated concisely in terms of an inverse with respect to the semi-inner product induced by the interpolation kernel. Based on this representation, a separation of the solution process is justified and consequently splitting methods and an orthogonal projection method based on the semi-inner norm induced by the RBF are established. The requirements for preconditioning operators are derived and exemplary domain decomposition method preconditioning operators are presented. The introduced representation using the inverse with respect to the semi-inner product clarifies the coherence with well-known concepts from numerical linear algebra. The generic formulation of the preconditioned orthogonal projection method and the requirements for suitable preconditioners serve as building blocks to create solvers tailored for the specific assets of available hardware. Exemplary, design variants of the established subspace projection method and the respective preconditioners are tested on replicable data up to interpolation centers.
{"title":"Generic preconditioning based on the inverse with respect to the semi-inner product for iterative solvers for radial basis function interpolation","authors":"Dirk Martin , Gundolf Haase , Günter Offner","doi":"10.1016/j.cam.2024.116355","DOIUrl":"10.1016/j.cam.2024.116355","url":null,"abstract":"<div><div>The inverse matrix of radial basis function (RBF) interpolation systems can be stated concisely in terms of an inverse with respect to the semi-inner product induced by the interpolation kernel. Based on this representation, a separation of the solution process is justified and consequently splitting methods and an orthogonal projection method based on the semi-inner norm induced by the RBF are established. The requirements for preconditioning operators are derived and exemplary domain decomposition method preconditioning operators are presented. The introduced representation using the inverse with respect to the semi-inner product clarifies the coherence with well-known concepts from numerical linear algebra. The generic formulation of the preconditioned orthogonal projection method and the requirements for suitable preconditioners serve as building blocks to create solvers tailored for the specific assets of available hardware. Exemplary, design variants of the established subspace projection method and the respective preconditioners are tested on replicable data up to <span><math><msup><mrow><mn>2</mn></mrow><mrow><mn>19</mn></mrow></msup></math></span> interpolation centers.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"459 ","pages":"Article 116355"},"PeriodicalIF":2.1,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661489","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 : 2024-11-09DOI: 10.1016/j.cam.2024.116375
Zhijun Tan , Yunhua Zeng
In this paper, we have developed a temporal second-order two-grid FEM to solve the semilinear time-fractional Rayleigh–Stokes equations. The proposed two-grid FEM uses the L2- scheme and second order scheme to approximate the Caputo fractional derivative and the time first-order derivative in temporal direction and the standard FEM in spatial direction. The -norm and -norm stability and error estimates for the standard finite element solution and the two-grid solution are derived. The results shown that as long as the mesh sizes satisfy and respectively, the two-grid algorithm can achieve asymptotically optimal approximation. Furthermore, the non-uniform L2- scheme was applied for temporal discretization to handle the weak singularity of the solution. Finally, the theoretical findings were confirmed by numerical results, and the effectiveness of the two-grid algorithm was demonstrated.
{"title":"Temporal second-order two-grid finite element method for semilinear time-fractional Rayleigh–Stokes equations","authors":"Zhijun Tan , Yunhua Zeng","doi":"10.1016/j.cam.2024.116375","DOIUrl":"10.1016/j.cam.2024.116375","url":null,"abstract":"<div><div>In this paper, we have developed a temporal second-order two-grid FEM to solve the semilinear time-fractional Rayleigh–Stokes equations. The proposed two-grid FEM uses the L2-<span><math><msub><mrow><mn>1</mn></mrow><mrow><mi>σ</mi></mrow></msub></math></span> scheme and second order scheme to approximate the Caputo fractional derivative and the time first-order derivative in temporal direction and the standard FEM in spatial direction. The <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm stability and error estimates for the standard finite element solution and the two-grid solution are derived. The results shown that as long as the mesh sizes satisfy <span><math><mrow><mi>H</mi><mo>=</mo><msup><mrow><mi>h</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></math></span> and <span><math><mrow><mi>H</mi><mo>=</mo><msup><mrow><mi>h</mi></mrow><mrow><mfrac><mrow><mi>r</mi></mrow><mrow><mn>2</mn><mi>r</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></msup></mrow></math></span> respectively, the two-grid algorithm can achieve asymptotically optimal approximation. Furthermore, the non-uniform L2-<span><math><msub><mrow><mn>1</mn></mrow><mrow><mi>σ</mi></mrow></msub></math></span> scheme was applied for temporal discretization to handle the weak singularity of the solution. Finally, the theoretical findings were confirmed by numerical results, and the effectiveness of the two-grid algorithm was demonstrated.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"459 ","pages":"Article 116375"},"PeriodicalIF":2.1,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660934","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 : 2024-11-09DOI: 10.1016/j.cam.2024.116361
Pin-Bo Chen , Gui-Hua Lin , Zhen-Ping Yang
In this paper, we study a stochastic nonsmooth second-order cone complementarity problem (SNS-SOCCP), in which the mathematical expectations are involved and the function is locally Lipschitz continuous but not necessarily continuously differentiable everywhere. By using some second-order cone complementarity function, SNS-SOCCP is reformulated equivalently into a system of stochastic nonsmooth equations. Based on this reformulation, we derive an explicit generalized Jacobian involved. Then, we design an inexact semismooth Newton algorithm based on an SAA (sample average approximation) technique to solve the stochastic nonsmooth equations. We investigate the convergence properties of the proposed algorithm under suitable conditions. Finally, to prove the effectiveness of the proposed algorithm, we solve numerically a stochastic power flow programming problem.
{"title":"An inexact semismooth Newton SAA-based algorithm for stochastic nonsmooth SOC complementarity problems with application to a stochastic power flow programming problem","authors":"Pin-Bo Chen , Gui-Hua Lin , Zhen-Ping Yang","doi":"10.1016/j.cam.2024.116361","DOIUrl":"10.1016/j.cam.2024.116361","url":null,"abstract":"<div><div>In this paper, we study a stochastic nonsmooth second-order cone complementarity problem (SNS-SOCCP), in which the mathematical expectations are involved and the function is locally Lipschitz continuous but not necessarily continuously differentiable everywhere. By using some second-order cone complementarity function, SNS-SOCCP is reformulated equivalently into a system of stochastic nonsmooth equations. Based on this reformulation, we derive an explicit generalized Jacobian involved. Then, we design an inexact semismooth Newton algorithm based on an SAA (sample average approximation) technique to solve the stochastic nonsmooth equations. We investigate the convergence properties of the proposed algorithm under suitable conditions. Finally, to prove the effectiveness of the proposed algorithm, we solve numerically a stochastic power flow programming problem.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"458 ","pages":"Article 116361"},"PeriodicalIF":2.1,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652788","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 : 2024-11-09DOI: 10.1016/j.cam.2024.116368
Xiaoqing Meng , Aijie Cheng , Zhengguang Liu
We propose an energy-optimized invariant energy quadratization method to solve the gradient flow models in this paper, which requires only one linear energy-optimized step to correct the auxiliary variables on each time step. In addition to inheriting the benefits of the baseline and relaxed invariant energy quadratization method, our approach has several other advantages. Firstly, in the process of correcting auxiliary variables, we can directly solve linear programming problem by the energy-optimized technique, which greatly simplifies the nonlinear optimization problem in the previous relaxed invariant energy quadratization method. Secondly, we construct new linear unconditionally energy stable schemes by applying backward differentiation formulas and Crank–Nicolson formula, so that the accuracy in time can reach the first- and second-order. Thirdly, comparing with relaxation technique, the modified energy obtained by energy-optimized technique is closer to the original energy, and the accuracy and consistency of the numerical solutions can be improved. Ample numerical examples have been presented to demonstrate the accuracy, efficiency and energy stability of the proposed schemes.
{"title":"Improving the accuracy and consistency of the energy quadratization method with an energy-optimized technique","authors":"Xiaoqing Meng , Aijie Cheng , Zhengguang Liu","doi":"10.1016/j.cam.2024.116368","DOIUrl":"10.1016/j.cam.2024.116368","url":null,"abstract":"<div><div>We propose an energy-optimized invariant energy quadratization method to solve the gradient flow models in this paper, which requires only one linear energy-optimized step to correct the auxiliary variables on each time step. In addition to inheriting the benefits of the baseline and relaxed invariant energy quadratization method, our approach has several other advantages. Firstly, in the process of correcting auxiliary variables, we can directly solve linear programming problem by the energy-optimized technique, which greatly simplifies the nonlinear optimization problem in the previous relaxed invariant energy quadratization method. Secondly, we construct new linear unconditionally energy stable schemes by applying backward differentiation formulas and Crank–Nicolson formula, so that the accuracy in time can reach the first- and second-order. Thirdly, comparing with relaxation technique, the modified energy obtained by energy-optimized technique is closer to the original energy, and the accuracy and consistency of the numerical solutions can be improved. Ample numerical examples have been presented to demonstrate the accuracy, efficiency and energy stability of the proposed schemes.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"459 ","pages":"Article 116368"},"PeriodicalIF":2.1,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661491","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 : 2024-11-09DOI: 10.1016/j.cam.2024.116369
Xiaohui Wu , Yanping Chen , Yang Wang
In this paper, we present a two-grid virtual element method to solve the nonlinear parabolic problem. The nonlinear terms are approximated by using the orthogonal projection, and the fine-grid discrete form is enhanced by Newton iteration. We first prove the -norm error estimate for the fully discrete problem. Furthermore, the a priori error estimates of two-grid method in the - and -norms achieve the optimal order and , respectively. Finally, we used two numerical examples to validate our two-grid algorithm, which is consistent with our theoretical results.
{"title":"Error analysis of two-grid virtual element method for nonlinear parabolic problems on general polygonal meshes","authors":"Xiaohui Wu , Yanping Chen , Yang Wang","doi":"10.1016/j.cam.2024.116369","DOIUrl":"10.1016/j.cam.2024.116369","url":null,"abstract":"<div><div>In this paper, we present a two-grid virtual element method to solve the nonlinear parabolic problem. The nonlinear terms <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span> are approximated by using the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> orthogonal projection, and the fine-grid discrete form is enhanced by Newton iteration. We first prove the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm error estimate for the fully discrete problem. Furthermore, the a priori error estimates of two-grid method in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>- and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norms achieve the optimal order <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msup><mo>+</mo><mi>τ</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>+</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msup><mo>+</mo><mi>τ</mi><mo>)</mo></mrow></mrow></math></span>, respectively. Finally, we used two numerical examples to validate our two-grid algorithm, which is consistent with our theoretical results.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"459 ","pages":"Article 116369"},"PeriodicalIF":2.1,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661490","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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