In this paper, we consider 3D steady convection-diffusion equations in cylindrical domains. Instead of applying the finite difference methods (FDM) or the finite element methods (FEM), we propose a difference finite element method (DFEM) that can maximize good applicability and efficiency of both FDM and FEM. The essence of this method lies in employing the centered difference discretization in the $z$-direction and the finite element discretization based on the $P_1$ conforming elements in the $(x, y)$ plane. This allows us to solve partial differential equations on complex cylindrical domains at lower computational costs compared to applying the 3D finite element method. We derive stability estimates for the difference finite element solution and establish the explicit dependence of $H_1$ error bounds on the diffusivity, convection field modulus, and mesh size. Finally, we provide numerical examples to verify the theoretical predictions and showcase the accuracy of the considered method.
{"title":"A Difference Finite Element Method for Convection-Diffusion Equations in Cylindrical Domains","authors":"Chenhong Shi,Yinnian He,Dongwoo Sheen, Xinlong Feng","doi":"10.4208/ijnam2024-1016","DOIUrl":"https://doi.org/10.4208/ijnam2024-1016","url":null,"abstract":"In this paper, we consider 3D steady convection-diffusion equations in cylindrical\u0000domains. Instead of applying the finite difference methods (FDM) or the finite element methods\u0000(FEM), we propose a difference finite element method (DFEM) that can maximize good applicability and efficiency of both FDM and FEM. The essence of this method lies in employing the\u0000centered difference discretization in the $z$-direction and the finite element discretization based on\u0000the $P_1$ conforming elements in the $(x, y)$ plane. This allows us to solve partial differential equations on complex cylindrical domains at lower computational costs compared to applying the 3D\u0000finite element method. We derive stability estimates for the difference finite element solution and\u0000establish the explicit dependence of $H_1$ error bounds on the diffusivity, convection field modulus,\u0000and mesh size. Finally, we provide numerical examples to verify the theoretical predictions and\u0000showcase the accuracy of the considered method.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"40 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141150441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, by introducing a new flux variable, two decoupled and linearized block-centered finite difference methods are developed and analyzed for the nonlinear symmetric regularized long wave equation, where the two-step backward difference formula and Crank-Nicolson temporal discretization combined with linear extrapolation technique are employed. Under a reasonable time stepsize ratio restriction, i.e., $∆t=o(h^{1/4}),$ second-order convergence for both the primal variable and its flux are rigorously proved on general non-uniform spatial grids. Moreover, based upon the convergence results and inverse estimate, stability of two methods are also demonstrated. Ample numerical experiments are presented to confirm the theoretical analysis.
{"title":"Two Decoupled and Linearized Block-Centered Finite Difference Methods for the Nonlinear Symmetric Regularized Long Wave Equation","authors":"Jie Xu,Shusen Xie, Hongfei Fu","doi":"10.4208/ijnam2024-1010","DOIUrl":"https://doi.org/10.4208/ijnam2024-1010","url":null,"abstract":"In this paper, by introducing a new flux variable, two decoupled and linearized block-centered finite difference methods are developed and analyzed for the nonlinear symmetric regularized long wave equation, where the two-step backward difference formula and Crank-Nicolson\u0000temporal discretization combined with linear extrapolation technique are employed. Under a reasonable time stepsize ratio restriction, i.e., $∆t=o(h^{1/4}),$ second-order convergence for both the\u0000primal variable and its flux are rigorously proved on general non-uniform spatial grids. Moreover, based upon the convergence results and inverse estimate, stability of two methods are also\u0000demonstrated. Ample numerical experiments are presented to confirm the theoretical analysis.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"71 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140598670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a semi-discrete finite element method for a dynamic model for linear viscoelastic materials based on the constitutive law of fractional order. The corresponding integro-differential equation is of a Mittag-Leffler type convolution kernel. A 4-node hybrid stress quadrilateral finite element is used for the spatial discretization. We show the existence and uniqueness of the semi-discrete solution, then derive some error estimates. Finally, we provide several numerical examples to verify the theoretical results.
{"title":"A Hybrid Stress Finite Element Method for Integro-Differential Equations Modelling Dynamic Fractional Order Viscoelasticity","authors":"Menghan Liu, Xiaoping Xie","doi":"10.4208/ijnam2024-1009","DOIUrl":"https://doi.org/10.4208/ijnam2024-1009","url":null,"abstract":"We consider a semi-discrete finite element method for a dynamic model for linear viscoelastic materials based on the constitutive law of fractional order. The corresponding\u0000integro-differential equation is of a Mittag-Leffler type convolution kernel. A 4-node hybrid stress\u0000quadrilateral finite element is used for the spatial discretization. We show the existence and\u0000uniqueness of the semi-discrete solution, then derive some error estimates. Finally, we provide\u0000several numerical examples to verify the theoretical results.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"18 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140598622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A solenoidal basis is constructed to compute velocities using a certain finite element method for the Stokes problem. The method is conforming, with piecewise linear velocity and piecewise constant pressure on the Powell-Sabin split of a triangulation. Inhomogeneous Dirichlet conditions are supported by constructing an interpolating operator into the solenoidal velocity space. The solenoidal basis reduces the problem size and eliminates the pressure variable from the linear system for the velocity. A basis of the pressure space is also constructed that can be used to compute the pressure after the velocity, if it is desired to compute the pressure. All basis functions have local support and lead to sparse linear systems. The basis construction is confirmed through rigorous analysis. Velocity and pressure system matrices are both symmetric, positive definite, which can be exploited to solve their corresponding linear systems. Significant efficiency gains over the usual saddle-point formulation are demonstrated computationally.
{"title":"An $H^1$-Conforming Solenoidal Basis for Velocity Computation on Powell-Sabin Splits for the Stokes Problem","authors":"Jeffrey M. Connors, Michael Gaiewski","doi":"10.4208/ijnam2024-1007","DOIUrl":"https://doi.org/10.4208/ijnam2024-1007","url":null,"abstract":"A solenoidal basis is constructed to compute velocities using a certain finite element\u0000method for the Stokes problem. The method is conforming, with piecewise linear velocity and\u0000piecewise constant pressure on the Powell-Sabin split of a triangulation. Inhomogeneous Dirichlet\u0000conditions are supported by constructing an interpolating operator into the solenoidal velocity\u0000space. The solenoidal basis reduces the problem size and eliminates the pressure variable from the\u0000linear system for the velocity. A basis of the pressure space is also constructed that can be used to\u0000compute the pressure after the velocity, if it is desired to compute the pressure. All basis functions\u0000have local support and lead to sparse linear systems. The basis construction is confirmed through\u0000rigorous analysis. Velocity and pressure system matrices are both symmetric, positive definite,\u0000which can be exploited to solve their corresponding linear systems. Significant efficiency gains\u0000over the usual saddle-point formulation are demonstrated computationally.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"34 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140598779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An efficient iteration method is provided in this paper for solving a class of nonlinear systems whose Jacobian matrices are complex and symmetric. The modified Newton-NDSS method is developed and applied to the class of nonlinear systems by adopting the modified Newton method as the outer solver and a new double-step splitting (NDSS) iteration scheme as the inner solver. Additionally, we theoretically analyze the local convergent properties of the new method under the weaker Hölder conditions. Lastly, the new method is compared numerically with some existing ones and the numerical experiments solving the nonlinear equations demonstrate the superiority of the Newton-NDSS method.
{"title":"Modified Newton-NDSS Method for Solving Nonlinear System with Complex Symmetric Jacobian Matrices","authors":"Xiaohui Yu, Qingbiao Wu","doi":"10.4208/ijnam2024-1012","DOIUrl":"https://doi.org/10.4208/ijnam2024-1012","url":null,"abstract":"An efficient iteration method is provided in this paper for solving a class of nonlinear systems whose Jacobian matrices are complex and symmetric. The modified Newton-NDSS\u0000method is developed and applied to the class of nonlinear systems by adopting the modified\u0000Newton method as the outer solver and a new double-step splitting (NDSS) iteration scheme as\u0000the inner solver. Additionally, we theoretically analyze the local convergent properties of the new\u0000method under the weaker Hölder conditions. Lastly, the new method is compared numerically with\u0000some existing ones and the numerical experiments solving the nonlinear equations demonstrate\u0000the superiority of the Newton-NDSS method.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"103 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140598625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we consider Richardson extrapolation of the Crank-Nicolson (CN) scheme for backward stochastic differential equations (BSDEs). First, applying the Adomian decomposition to the nonlinear generator of BSDEs, we introduce a new system of BSDEs. Then we theoretically prove that the solution of the CN scheme for BSDEs admits an asymptotic expansion with its coefficients the solutions of the new system of BSDEs. Based on the expansion, we propose Richardson extrapolation algorithms for solving BSDEs. Finally, some numerical tests are carried out to verify our theoretical conclusions and to show the stability, efficiency and high accuracy of the algorithms.
{"title":"Richardson Extrapolation of the Crank-Nicolson Scheme for Backward Stochastic Differential Equations","authors":"Yafei Xu, Weidong Zhao","doi":"10.4208/ijnam2024-1011","DOIUrl":"https://doi.org/10.4208/ijnam2024-1011","url":null,"abstract":"In this work, we consider Richardson extrapolation of the Crank-Nicolson (CN)\u0000scheme for backward stochastic differential equations (BSDEs). First, applying the Adomian\u0000decomposition to the nonlinear generator of BSDEs, we introduce a new system of BSDEs. Then\u0000we theoretically prove that the solution of the CN scheme for BSDEs admits an asymptotic\u0000expansion with its coefficients the solutions of the new system of BSDEs. Based on the expansion,\u0000we propose Richardson extrapolation algorithms for solving BSDEs. Finally, some numerical tests\u0000are carried out to verify our theoretical conclusions and to show the stability, efficiency and high\u0000accuracy of the algorithms.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"100 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140598792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Noelia Bazarra,José R. Fernández,Antonio Magaña,Marc Magaña, Ramόn Quintanilla
In this work, we study a strain gradient problem involving mixtures. The variational formulation is written as a first-order in time coupled system of parabolic variational equations. An existence and uniqueness result is recalled. Then, we introduce a fully discrete approximation by using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved. Finally, some one- and two-dimensional numerical simulations are performed.
{"title":"An a Priori Error Analysis of a Problem Involving Mixtures of Continua with Gradient Enrichment","authors":"Noelia Bazarra,José R. Fernández,Antonio Magaña,Marc Magaña, Ramόn Quintanilla","doi":"10.4208/ijnam2024-1006","DOIUrl":"https://doi.org/10.4208/ijnam2024-1006","url":null,"abstract":"In this work, we study a strain gradient problem involving mixtures. The variational\u0000formulation is written as a first-order in time coupled system of parabolic variational equations.\u0000An existence and uniqueness result is recalled. Then, we introduce a fully discrete approximation\u0000by using the finite element method and the implicit Euler scheme. A discrete stability property and\u0000a priori error estimates are proved. Finally, some one- and two-dimensional numerical simulations\u0000are performed.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"13 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140598621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to a discontinuous Galerkin (DG) method for nonlinear quasi-static poroelasticity problems. The fully implicit nonlinear numerical scheme is constructed by utilizing DG method for the spatial approximation and the backward Euler method for the temporal discretization. The existence and uniqueness of the numerical solution is proved. Then we derive the optimal convergence order estimates in a discrete $H^1$ norm for the displacement and in $H^1$ and $L^2$ norms for the pressure. Finally, numerical experiments are supplied to validate the theoretical error estimates of our proposed method.
{"title":"Discontinuous Galerkin Method for Nonlinear Quasi-Static Poroelasticity Problems","authors":"Fan Chen,Ming Cui, Chenguang Zhou","doi":"10.4208/ijnam2024-1008","DOIUrl":"https://doi.org/10.4208/ijnam2024-1008","url":null,"abstract":"This paper is devoted to a discontinuous Galerkin (DG) method for nonlinear quasi-static poroelasticity problems. The fully implicit nonlinear numerical scheme is constructed by\u0000utilizing DG method for the spatial approximation and the backward Euler method for the temporal discretization. The existence and uniqueness of the numerical solution is proved. Then we\u0000derive the optimal convergence order estimates in a discrete $H^1$ norm for the displacement and\u0000in $H^1$ and $L^2$ norms for the pressure. Finally, numerical experiments are supplied to validate the\u0000theoretical error estimates of our proposed method.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"13 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140598906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider mixed finite element approximation of a coupled system of nonlinear parabolic advection-diffusion-reaction variational (in)equalities modeling biofilm growth and nutrient utilization in porous media at pore-scale. We study well-posedness of the discrete system and derive an optimal error estimate of first order. Our theoretical estimates extend the work on a scalar degenerate parabolic problem by Arbogast et al, 1997 [4] to a variational inequality; we also apply it to a system. We also verify our theoretical convergence results with simulations of realistic scenarios.
{"title":"Numerical Analysis of a Mixed Finite Element Approximation of a Coupled System Modeling Biofilm Growth in Porous Media with Simulations","authors":"Azhar Alhammali,Malgorzata Peszynska, Choah Shin","doi":"10.4208/ijnam2024-1002","DOIUrl":"https://doi.org/10.4208/ijnam2024-1002","url":null,"abstract":"In this paper, we consider mixed finite element approximation of a coupled system of\u0000nonlinear parabolic advection-diffusion-reaction variational (in)equalities modeling biofilm growth\u0000and nutrient utilization in porous media at pore-scale. We study well-posedness of the discrete\u0000system and derive an optimal error estimate of first order. Our theoretical estimates extend\u0000the work on a scalar degenerate parabolic problem by Arbogast et al, 1997 [4] to a variational\u0000inequality; we also apply it to a system. We also verify our theoretical convergence results with\u0000simulations of realistic scenarios.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"111 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139083355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A parallelizable iterative procedure based on domain decomposition is presented and analyzed for weak Galerkin finite element methods for second order elliptic equations. The convergence analysis is established for the decomposition of the domain into individual elements associated to the weak Galerkin methods or into larger subdomains. A series of numerical tests are illustrated to verify the theory developed in this paper.
{"title":"A Parallel Iterative Procedure for Weak Galerkin Methods for Second Order Elliptic Problems","authors":"Chunmei Wang,Junping Wang, Shangyou Zhang","doi":"10.4208/ijnam2024-1001","DOIUrl":"https://doi.org/10.4208/ijnam2024-1001","url":null,"abstract":"A parallelizable iterative procedure based on domain decomposition is presented\u0000and analyzed for weak Galerkin finite element methods for second order elliptic equations. The\u0000convergence analysis is established for the decomposition of the domain into individual elements\u0000associated to the weak Galerkin methods or into larger subdomains. A series of numerical tests\u0000are illustrated to verify the theory developed in this paper.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"14 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139084183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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