Pub Date : 2024-11-13DOI: 10.1016/j.apnum.2024.11.002
Zhihao Ge , Yanan He
In this paper, we propose a new multiphysics finite element method for a quasi-static poroelasticity model. Firstly, to overcome the displacement locking phenomenon and pressure oscillation, we reformulate the original model into a fluid-fluid coupling problem by introducing new variables-the generalized nonlocal Stokes equations and a diffusion equation, which is a completely new model. Then, we design a fully discrete multiphysics finite element method for the reformulated model-linear finite element pairs for the spatial variables and backward Euler method for time discretization. And we prove that the proposed method is stable without any stabilized term and robust for many parameters and it has the optimal convergence order. Finally, we show some numerical tests to verify the theoretical results.
{"title":"A new multiphysics finite element method for a quasi-static poroelasticity model","authors":"Zhihao Ge , Yanan He","doi":"10.1016/j.apnum.2024.11.002","DOIUrl":"10.1016/j.apnum.2024.11.002","url":null,"abstract":"<div><div>In this paper, we propose a new multiphysics finite element method for a quasi-static poroelasticity model. Firstly, to overcome the displacement locking phenomenon and pressure oscillation, we reformulate the original model into a fluid-fluid coupling problem by introducing new variables-the generalized nonlocal Stokes equations and a diffusion equation, which is a completely new model. Then, we design a fully discrete multiphysics finite element method for the reformulated model-linear finite element pairs for the spatial variables <span><math><mo>(</mo><mi>u</mi><mo>,</mo><mi>ξ</mi><mo>,</mo><mi>η</mi><mo>)</mo></math></span> and backward Euler method for time discretization. And we prove that the proposed method is stable without any stabilized term and robust for many parameters and it has the optimal convergence order. Finally, we show some numerical tests to verify the theoretical results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"209 ","pages":"Pages 1-19"},"PeriodicalIF":2.2,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142653222","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 : 2024-10-05DOI: 10.1016/j.apnum.2024.10.001
Constantino Caetano , Luísa Morgado , Pedro Lima , Niel Hens , Baltazar Nunes
This study introduces a SIR (Susceptible-Infectious-Recovered) model using fractional derivatives to assess the population's hesitancy to the COVID-19 vaccination campaign in Portugal. Leveraging the framework developed by Angstmann [1], our approach incorporates fractional derivatives to best describe the nuanced dynamics of the vaccination process. We begin by examining the qualitative properties of the proposed model. To substantiate the inclusion of fractional derivatives, empirical data along with statistical criteria are applied. Numerical simulations are performed to compare both integer and fractional order models. An epidemiological interpretation for the fractional order of the model is provided, in the context of a vaccination campaign.
{"title":"A fractional order SIR model describing hesitancy to the COVID-19 vaccination","authors":"Constantino Caetano , Luísa Morgado , Pedro Lima , Niel Hens , Baltazar Nunes","doi":"10.1016/j.apnum.2024.10.001","DOIUrl":"10.1016/j.apnum.2024.10.001","url":null,"abstract":"<div><div>This study introduces a SIR (Susceptible-Infectious-Recovered) model using fractional derivatives to assess the population's hesitancy to the COVID-19 vaccination campaign in Portugal. Leveraging the framework developed by Angstmann <span><span>[1]</span></span>, our approach incorporates fractional derivatives to best describe the nuanced dynamics of the vaccination process. We begin by examining the qualitative properties of the proposed model. To substantiate the inclusion of fractional derivatives, empirical data along with statistical criteria are applied. Numerical simulations are performed to compare both integer and fractional order models. An epidemiological interpretation for the fractional order of the model is provided, in the context of a vaccination campaign.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"207 ","pages":"Pages 608-620"},"PeriodicalIF":2.2,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421774","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 : 2024-10-04DOI: 10.1016/j.apnum.2024.09.029
Kai Jiang, Shifeng Li, Juan Zhang
The complex continuous-time algebraic Riccati equation (CCARE) is quadratic, which is closely related to the analysis of the optimal control problem. In this paper, we apply Newton method as the outer iteration and an efficient general alternating-direction implicit (GADI) method as the inner iteration to solve CCARE. Meanwhile, we propose the inexact Newton-GADI method to further improve the efficiency of the algorithm. We give the convergence analysis of our proposed method and prove that its convergence rate is faster than the classical Newton-ADI method. Finally, some numerical examples are given to illustrate the effectiveness of our algorithms and the correctness of the theoretical analysis.
{"title":"A general alternating-direction implicit Newton method for solving continuous-time algebraic Riccati equation","authors":"Kai Jiang, Shifeng Li, Juan Zhang","doi":"10.1016/j.apnum.2024.09.029","DOIUrl":"10.1016/j.apnum.2024.09.029","url":null,"abstract":"<div><div>The complex continuous-time algebraic Riccati equation (CCARE) is quadratic, which is closely related to the analysis of the optimal control problem. In this paper, we apply Newton method as the outer iteration and an efficient general alternating-direction implicit (GADI) method as the inner iteration to solve CCARE. Meanwhile, we propose the inexact Newton-GADI method to further improve the efficiency of the algorithm. We give the convergence analysis of our proposed method and prove that its convergence rate is faster than the classical Newton-ADI method. Finally, some numerical examples are given to illustrate the effectiveness of our algorithms and the correctness of the theoretical analysis.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"207 ","pages":"Pages 642-656"},"PeriodicalIF":2.2,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421773","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 : 2024-10-03DOI: 10.1016/j.apnum.2024.09.028
Lixiang Jin, Zhaoxiang Li, Peipei Wang, Lijun Yi
In this paper, we develop two numerical methods, the Legendre-Galerkin method and the generalized Log orthogonal functions Galerkin method for numerically solving the fully nonlinear Monge-Ampère equation. Both methods are constructed based on the vanishing moment approach. To address both solution stability and computational efficiency, we propose a multiple-level framework for resolving discretization schemes. The mathematical justifications of the new approaches and the error estimates for the Legendre-Galerkin method are established. Numerical experiments validate the accuracy of our methods, and a comparative experiment demonstrates the advantage of Log orthogonal functions for problems with corner singularities. The results highlight that our methods have high-order accuracy and small computational cost.
{"title":"Spectral-Galerkin methods for the fully nonlinear Monge-Ampère equation","authors":"Lixiang Jin, Zhaoxiang Li, Peipei Wang, Lijun Yi","doi":"10.1016/j.apnum.2024.09.028","DOIUrl":"10.1016/j.apnum.2024.09.028","url":null,"abstract":"<div><div>In this paper, we develop two numerical methods, the Legendre-Galerkin method and the generalized Log orthogonal functions Galerkin method for numerically solving the fully nonlinear Monge-Ampère equation. Both methods are constructed based on the vanishing moment approach. To address both solution stability and computational efficiency, we propose a multiple-level framework for resolving discretization schemes. The mathematical justifications of the new approaches and the error estimates for the Legendre-Galerkin method are established. Numerical experiments validate the accuracy of our methods, and a comparative experiment demonstrates the advantage of Log orthogonal functions for problems with corner singularities. The results highlight that our methods have high-order accuracy and small computational cost.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"207 ","pages":"Pages 621-641"},"PeriodicalIF":2.2,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421772","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 : 2024-10-01DOI: 10.1016/j.apnum.2024.09.027
Xiaojuan Shen, Yongyong Cai
In this paper, we present a rigorous error analysis for two weakly decoupled, unconditionally energy stable schemes in the semi-discrete-in-time form. The methods consist of a stabilized/convex-splitting method for the phase field equations and a projection correction method for the MHD model. Several numerical simulations demonstrate the validity of theoretical results.
{"title":"Error estimates of time discretizations for a Cahn-Hilliard phase-field model for the two-phase magnetohydrodynamic flows","authors":"Xiaojuan Shen, Yongyong Cai","doi":"10.1016/j.apnum.2024.09.027","DOIUrl":"10.1016/j.apnum.2024.09.027","url":null,"abstract":"<div><div>In this paper, we present a rigorous error analysis for two weakly decoupled, unconditionally energy stable schemes in the semi-discrete-in-time form. The methods consist of a stabilized/convex-splitting method for the phase field equations and a projection correction method for the MHD model. Several numerical simulations demonstrate the validity of theoretical results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"207 ","pages":"Pages 585-607"},"PeriodicalIF":2.2,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421776","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 : 2024-09-25DOI: 10.1016/j.apnum.2024.09.020
Shridhar Kumar, Pratibhamoy Das
This study addresses time-dependent multiple-scale reaction-convection-diffusion initial boundary value systems characterized by strong coupling in the reaction matrix and weak coupling in the convection terms for a locally optimal accurate solution. The discrete problem, which typically loses its tridiagonal structure, expands its bandwidth to four in such coupled systems, resulting in a substantial computational load. Our objective is to mitigate this computational burden through a splitting approach that transforms the non-tridiagonal matrix into a tridiagonal form while maintaining the consistency, local optimal accuracy in space, and stability of the numerical scheme. We employ equidistributed non-uniform grids, guided by a carefully chosen monitor function, to approximate the continuous space domain. The discretization strategy targets local optimal linear accuracy across space and time on the domain's interior points. In addition, we have also provided the global convergence analysis of the present splitting approach, mathematically. The mathematical evidence is also obtained from the numerical experiments by comparing the splitting approach (either diagonal or triangular forms) of the reaction matrix to its coupled form. The results strongly confirm the effectiveness of this approach in delivering uniform linear accuracy, based on the present problem discretizations while significantly reducing the computational costs.
{"title":"A uniformly convergent analysis for multiple scale parabolic singularly perturbed convection-diffusion coupled systems: Optimal accuracy with less computational time","authors":"Shridhar Kumar, Pratibhamoy Das","doi":"10.1016/j.apnum.2024.09.020","DOIUrl":"10.1016/j.apnum.2024.09.020","url":null,"abstract":"<div><div>This study addresses time-dependent multiple-scale reaction-convection-diffusion initial boundary value systems characterized by strong coupling in the reaction matrix and weak coupling in the convection terms for a locally optimal accurate solution. The discrete problem, which typically loses its tridiagonal structure, expands its bandwidth to four in such coupled systems, resulting in a substantial computational load. Our objective is to mitigate this computational burden through a splitting approach that transforms the non-tridiagonal matrix into a tridiagonal form while maintaining the consistency, local optimal accuracy in space, and stability of the numerical scheme. We employ equidistributed non-uniform grids, guided by a carefully chosen monitor function, to approximate the continuous space domain. The discretization strategy targets local optimal linear accuracy across space and time on the domain's interior points. In addition, we have also provided the global convergence analysis of the present splitting approach, mathematically. The mathematical evidence is also obtained from the numerical experiments by comparing the splitting approach (either diagonal or triangular forms) of the reaction matrix to its coupled form. The results strongly confirm the effectiveness of this approach in delivering uniform linear accuracy, based on the present problem discretizations while significantly reducing the computational costs.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"207 ","pages":"Pages 534-557"},"PeriodicalIF":2.2,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142322448","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 : 2024-09-23DOI: 10.1016/j.apnum.2024.09.018
Huailing Song , Qingkui Tan
In this paper, for the time-dependent parabolic equations defined on complex geometries domain, we develop and analyze the least-squares radial basis function finite difference method (RBF-FD) coupled with the implicit-explicit Runge-Kutta (IMEX-RK) time discretization up to third order accuracy, which improves stability and accuracy. We derive the absolute stability region and the optimal time-step constraint for four kinds of IMEX-RK schemes. Compared to the traditional explicit or implicit time discretization, these are not trivial. Under a wide time-step constraint, the stability and the error estimates in -norm are established. Finally, several numerical experiments on the regular domain and non-convex domain are performed to validate the theoretical analysis.
{"title":"Stability analysis and error estimates of implicit-explicit Runge-Kutta least squares RBF-FD method for time-dependent parabolic equation","authors":"Huailing Song , Qingkui Tan","doi":"10.1016/j.apnum.2024.09.018","DOIUrl":"10.1016/j.apnum.2024.09.018","url":null,"abstract":"<div><div>In this paper, for the time-dependent parabolic equations defined on complex geometries domain, we develop and analyze the least-squares radial basis function finite difference method (RBF-FD) coupled with the implicit-explicit Runge-Kutta (IMEX-RK) time discretization up to third order accuracy, which improves stability and accuracy. We derive the absolute stability region and the optimal time-step constraint for four kinds of IMEX-RK schemes. Compared to the traditional explicit or implicit time discretization, these are not trivial. Under a wide time-step constraint, the stability and the error estimates in <span><math><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-norm are established. Finally, several numerical experiments on the regular domain and non-convex domain are performed to validate the theoretical analysis.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"207 ","pages":"Pages 499-519"},"PeriodicalIF":2.2,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142322450","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 : 2024-09-23DOI: 10.1016/j.apnum.2024.09.019
Aayushman Raina, Srinivasan Natesan
In this paper, we propose a weak Galerkin finite element approximation for a class of fourth-order singularly perturbed parabolic problems. The problem exhibits boundary layers and so we have considered layer adapted triangulations, in particular Shishkin triangular mesh in the spatial domain. For temporal discretization, we utilize the Crank-Nicolson scheme on a uniform mesh. Stability and error estimates along with the uniform convergence of the method has been proved. Numerical examples are included which verifies our analysis.
{"title":"A weak Galerkin finite element method for fourth-order parabolic singularly perturbed problems on layer adapted Shishkin mesh","authors":"Aayushman Raina, Srinivasan Natesan","doi":"10.1016/j.apnum.2024.09.019","DOIUrl":"10.1016/j.apnum.2024.09.019","url":null,"abstract":"<div><div>In this paper, we propose a weak Galerkin finite element approximation for a class of fourth-order singularly perturbed parabolic problems. The problem exhibits boundary layers and so we have considered layer adapted triangulations, in particular Shishkin triangular mesh in the spatial domain. For temporal discretization, we utilize the Crank-Nicolson scheme on a uniform mesh. Stability and error estimates along with the uniform convergence of the method has been proved. Numerical examples are included which verifies our analysis.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"207 ","pages":"Pages 520-533"},"PeriodicalIF":2.2,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142322451","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 : 2024-09-19DOI: 10.1016/j.apnum.2024.09.017
Hui Peng , Wenya Qi
In this work, we develop a weak Galerkin method for the three-field Biot's consolidation model. The key idea is to consider the total pressure variable. We employ the stable pair of weak Galerkin finite elements to discretize the displacement and total pressure, and use totally discontinuous weak functions to approximate pressure in a semi-discrete scheme. Then, we give the fully discrete scheme based on the backward Euler method in time. Furthermore, we prove the well-posedness of the numerical schemes and derive the optimal error estimates for three variables in their nature norms. Our theoretical results are independent of the Lamé constant λ and the storage coefficient . Finally, some experiments that employ different polynomial degrees and polygonal meshes are presented to demonstrate the efficiency and stability of the weak Galerkin method.
{"title":"Weak Galerkin finite element method with the total pressure variable for Biot's consolidation model","authors":"Hui Peng , Wenya Qi","doi":"10.1016/j.apnum.2024.09.017","DOIUrl":"10.1016/j.apnum.2024.09.017","url":null,"abstract":"<div><div>In this work, we develop a weak Galerkin method for the three-field Biot's consolidation model. The key idea is to consider the total pressure variable. We employ the stable pair of weak Galerkin finite elements to discretize the displacement and total pressure, and use totally discontinuous weak functions to approximate pressure in a semi-discrete scheme. Then, we give the fully discrete scheme based on the backward Euler method in time. Furthermore, we prove the well-posedness of the numerical schemes and derive the optimal error estimates for three variables in their nature norms. Our theoretical results are independent of the Lamé constant <em>λ</em> and the storage coefficient <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. Finally, some experiments that employ different polynomial degrees and polygonal meshes are presented to demonstrate the efficiency and stability of the weak Galerkin method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"207 ","pages":"Pages 450-469"},"PeriodicalIF":2.2,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142313014","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 : 2024-09-19DOI: 10.1016/j.apnum.2024.09.016
Lei Xu, Li-Bin Liu, Zaitang Huang, Guangqing Long
In this paper, the nonsymmetric interior penalty Galerkin (NIPG) method on a Bakhvalov-type mesh is proposed for a singularly perturbed problem with two small parameters. In order to reflect the behavior of layers more accurately, a balanced norm, rather than the common energy norm, is introduced. By selecting special penalty parameters at different mesh points, we establish the supercloseness of order, and prove an optimal order of uniform convergence in a balanced norm. Numerical experiments are proposed to confirm our theoretical results.
{"title":"Supercloseness of the NIPG method on a Bakhvalov-type mesh for a singularly perturbed problem with two small parameters","authors":"Lei Xu, Li-Bin Liu, Zaitang Huang, Guangqing Long","doi":"10.1016/j.apnum.2024.09.016","DOIUrl":"10.1016/j.apnum.2024.09.016","url":null,"abstract":"<div><div>In this paper, the nonsymmetric interior penalty Galerkin (NIPG) method on a Bakhvalov-type mesh is proposed for a singularly perturbed problem with two small parameters. In order to reflect the behavior of layers more accurately, a balanced norm, rather than the common energy norm, is introduced. By selecting special penalty parameters at different mesh points, we establish the supercloseness of <span><math><mi>k</mi><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> order, and prove an optimal order of uniform convergence in a balanced norm. Numerical experiments are proposed to confirm our theoretical results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"207 ","pages":"Pages 431-449"},"PeriodicalIF":2.2,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142313013","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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