Pub Date : 2024-05-01DOI: 10.4208/nmtma.oa-2023-0124
Hailing Xuan,Xiaoliang Cheng, Xilu Wang
In this paper, we primarily investigate the existence, dependence and optimal control results related to solutions for a system of hemivariational inequalities pertaining to a non-stationary Navier-Stokes equation coupled with an evolution equation of temperature field. The boundary conditions for both the velocity field and temperature field incorporate the generalized Clarke gradient. The existence and uniqueness of the weak solution are established by utilizing the Banach fixed point theorem in conjunction with certain results pertaining to hemivariational inequalities. The finite element method is used to discretize the system of hemivariational inequalities and error bounds are derived. Ultimately, a result confirming the existence of a solution to an optimal control problem for the system of hemivariational inequalities is elucidated.
{"title":"Analysis and Optimal Control of a System of Hemivariational Inequalities Arising in Non-Stationary Navier-Stokes Equation with Thermal Effects","authors":"Hailing Xuan,Xiaoliang Cheng, Xilu Wang","doi":"10.4208/nmtma.oa-2023-0124","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2023-0124","url":null,"abstract":"In this paper, we primarily investigate the existence, dependence and optimal control results related to solutions for a system of hemivariational inequalities\u0000pertaining to a non-stationary Navier-Stokes equation coupled with an evolution\u0000equation of temperature field. The boundary conditions for both the velocity field\u0000and temperature field incorporate the generalized Clarke gradient. The existence\u0000and uniqueness of the weak solution are established by utilizing the Banach fixed\u0000point theorem in conjunction with certain results pertaining to hemivariational inequalities. The finite element method is used to discretize the system of hemivariational inequalities and error bounds are derived. Ultimately, a result confirming the\u0000existence of a solution to an optimal control problem for the system of hemivariational inequalities is elucidated.","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":"87 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925046","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}
Pub Date : 2024-05-01DOI: 10.4208/nmtma.oa-2023-0095
Siqing Li,Leevan Ling,Xin Liu,Pankaj K. Mishra,Mrinal K. Sen, Jing Zhang
Radial basis function generated finite-difference (RBF-FD) methods have recently gained popularity due to their flexibility with irregular node distributions. However, the convergence theories in the literature, when applied to nonuniform node distributions, require shrinking fill distance and do not take advantage of areas with high data density. Non-adaptive approach using same stencil size and degree of appended polynomial will have higher local accuracy at high density region, but has no effect on the overall order of convergence and could be a waste of computational power. This work proposes an adaptive RBF-FD method that utilizes the local data density to achieve a desirable order accuracy. By performing polynomial refinement and using adaptive stencil size based on data density, the adaptive RBFFD method yields differentiation matrices with higher sparsity while achieving the same user-specified convergence order for nonuniform point distributions. This allows the method to better leverage regions with higher node density, maintaining both accuracy and efficiency compared to standard non-adaptive RBF-FD methods.
{"title":"Enhancing RBF-FD Efficiency for Highly Non-Uniform Node Distributions via Adaptivity","authors":"Siqing Li,Leevan Ling,Xin Liu,Pankaj K. Mishra,Mrinal K. Sen, Jing Zhang","doi":"10.4208/nmtma.oa-2023-0095","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2023-0095","url":null,"abstract":"Radial basis function generated finite-difference (RBF-FD) methods have\u0000recently gained popularity due to their flexibility with irregular node distributions.\u0000However, the convergence theories in the literature, when applied to nonuniform\u0000node distributions, require shrinking fill distance and do not take advantage of areas\u0000with high data density. Non-adaptive approach using same stencil size and degree\u0000of appended polynomial will have higher local accuracy at high density region, but\u0000has no effect on the overall order of convergence and could be a waste of computational power. This work proposes an adaptive RBF-FD method that utilizes the\u0000local data density to achieve a desirable order accuracy. By performing polynomial\u0000refinement and using adaptive stencil size based on data density, the adaptive RBFFD method yields differentiation matrices with higher sparsity while achieving the\u0000same user-specified convergence order for nonuniform point distributions. This allows the method to better leverage regions with higher node density, maintaining\u0000both accuracy and efficiency compared to standard non-adaptive RBF-FD methods.","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":"8 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941711","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}
Pub Date : 2024-05-01DOI: 10.4208/nmtma.oa-2023-0163
Yiying Wang,Yongkui Zou,Xuan Liu, Chenguang Zhou
his paper presents error analysis of a stabilizer free weak Galerkin finite element method (SFWG-FEM) for second-order elliptic equations with low regularity solutions. The standard error analysis of SFWG-FEM requires additional regularity on solutions, such as $H^2$-regularity for the second-order convergence. However, if the solutions are in $H^{1+s}$ with $0 < s < 1,$ numerical experiments show that the SFWG-FEM is also effective and stable with the $(1+s)$-order convergence rate, so we develop a theoretical analysis for it. We introduce a standard $H^2$ finite element approximation for the elliptic problem, and then we apply the SFWG-FEM to approach this smooth approximating finite element solution. Finally, we establish the error analysis for SFWG-FEM with low regularity in both discrete $H^1$-norm and standard $L^2$-norm. The $(_Pk(T ), P_{k−1}(e), [P_{k+1}(T)]^d)$ elements with dimensions of space $d = 2, 3$ are employed and the numerical examples are tested to confirm the theory.
{"title":"A Stabilizer Free Weak Galerkin Finite Element Method for Elliptic Equation with Lower Regularity","authors":"Yiying Wang,Yongkui Zou,Xuan Liu, Chenguang Zhou","doi":"10.4208/nmtma.oa-2023-0163","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2023-0163","url":null,"abstract":"his paper presents error analysis of a stabilizer free weak Galerkin finite\u0000element method (SFWG-FEM) for second-order elliptic equations with low regularity solutions. The standard error analysis of SFWG-FEM requires additional regularity on solutions, such as $H^2$-regularity for the second-order convergence. However,\u0000if the solutions are in $H^{1+s}$ with $0 < s < 1,$ numerical experiments show that the\u0000SFWG-FEM is also effective and stable with the $(1+s)$-order convergence rate, so\u0000we develop a theoretical analysis for it. We introduce a standard $H^2$ finite element\u0000approximation for the elliptic problem, and then we apply the SFWG-FEM to approach this smooth approximating finite element solution. Finally, we establish the\u0000error analysis for SFWG-FEM with low regularity in both discrete $H^1$-norm and standard $L^2$-norm. The $(_Pk(T ), P_{k−1}(e), [P_{k+1}(T)]^d)$ elements with dimensions of space $d = 2, 3$ are employed and the numerical examples are tested to confirm the theory.","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":"146 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925053","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}
Pub Date : 2024-05-01DOI: 10.4208/nmtma.oa-2023-0122
Yaru Liu,Yinnian He, Xinlong Feng
In this paper, we propose an integral-averaged interpolation operator $I_tau$ in a bounded domain $Ω ⊂ mathbb{R}^n$ by using $Q_1$-element. The interpolation coefficient is defined by the average integral value of the interpolation function $u$ on the interval formed by the midpoints of the neighboring elements. The operator $I_tau$ reduces the regularity requirement for the function $u$ while maintaining standard convergence. Moreover, it possesses an important property of $||I_tau u||_{0,Ω} ≤ ||u||_{0,Ω}.$ We conduct stability analysis and error estimation for the operator $Itau.$ Finally, we present several numerical examples to test the efficiency and high accuracy of the operator
{"title":"Stability and Convergence of the Integral-Averaged Interpolation Operator Based on $Q_1$-element in $mathbb{R}^n$","authors":"Yaru Liu,Yinnian He, Xinlong Feng","doi":"10.4208/nmtma.oa-2023-0122","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2023-0122","url":null,"abstract":"In this paper, we propose an integral-averaged interpolation operator $I_tau$ in a bounded domain $Ω ⊂ mathbb{R}^n$ by using $Q_1$-element. The interpolation coefficient is\u0000defined by the average integral value of the interpolation function $u$ on the interval\u0000formed by the midpoints of the neighboring elements. The operator $I_tau$ reduces the\u0000regularity requirement for the function $u$ while maintaining standard convergence.\u0000Moreover, it possesses an important property of $||I_tau u||_{0,Ω} ≤ ||u||_{0,Ω}.$ We conduct\u0000stability analysis and error estimation for the operator $Itau.$ Finally, we present several\u0000numerical examples to test the efficiency and high accuracy of the operator","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":"10 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925131","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}
Pub Date : 2024-05-01DOI: 10.4208/nmtma.oa-2023-0136
Kai Gu,Peng Fang,Zhiwei Sun, Rui Du
We present a rigorous analysis of the convergence rate of the deep mixed residual method (MIM) when applied to a linear elliptic equation with different types of boundary conditions. The MIM has been proposed to solve high-order partial differential equations in high dimensions. Our analysis shows that MIM outperforms deep Ritz method and deep Galerkin method for weak solution in the Dirichlet case due to its ability to enforce the boundary condition. However, for the Neumann and Robin cases, MIM demonstrates similar performance to the other methods. Our results provide valuable insights into the strengths of MIM and its comparative performance in solving linear elliptic equations with different boundary conditions.
{"title":"Error Analysis of the Mixed Residual Method for Elliptic Equations","authors":"Kai Gu,Peng Fang,Zhiwei Sun, Rui Du","doi":"10.4208/nmtma.oa-2023-0136","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2023-0136","url":null,"abstract":"We present a rigorous analysis of the convergence rate of the deep mixed\u0000residual method (MIM) when applied to a linear elliptic equation with different\u0000types of boundary conditions. The MIM has been proposed to solve high-order partial differential equations in high dimensions. Our analysis shows that MIM outperforms deep Ritz method and deep Galerkin method for weak solution in the Dirichlet\u0000case due to its ability to enforce the boundary condition. However, for the Neumann\u0000and Robin cases, MIM demonstrates similar performance to the other methods. Our\u0000results provide valuable insights into the strengths of MIM and its comparative performance in solving linear elliptic equations with different boundary conditions.","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":"9 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925209","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}
Pub Date : 2024-02-01DOI: 10.4208/nmtma.oa-2023-0045
C.S. Chen,Andreas Karageorghis, Min Lei
We apply the local method of fundamental solutions (LMFS) to boundary value problems (BVPs) for the Laplace and homogeneous biharmonic equations in annuli. By appropriately choosing the collocation points, the LMFS discretization yields sparse block circulant system matrices. As a result, matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs) can be used for the solution of the systems resulting in considerable savings in both computational time and storage requirements. The accuracy of the method and its ability to solve large scale problems are demonstrated by applying it to several numerical experiments.
{"title":"Local MFS Matrix Decomposition Algorithms for Elliptic BVPs in Annuli","authors":"C.S. Chen,Andreas Karageorghis, Min Lei","doi":"10.4208/nmtma.oa-2023-0045","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2023-0045","url":null,"abstract":"We apply the local method of fundamental solutions (LMFS) to boundary\u0000value problems (BVPs) for the Laplace and homogeneous biharmonic equations in\u0000annuli. By appropriately choosing the collocation points, the LMFS discretization\u0000yields sparse block circulant system matrices. As a result, matrix decomposition\u0000algorithms (MDAs) and fast Fourier transforms (FFTs) can be used for the solution\u0000of the systems resulting in considerable savings in both computational time and\u0000storage requirements. The accuracy of the method and its ability to solve large scale\u0000problems are demonstrated by applying it to several numerical experiments.","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":"80 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139980108","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}
Pub Date : 2024-02-01DOI: 10.4208/nmtma.oa-2023-0058
Mo Chen,Yuling Jiao,Xiliang Lu,Pengcheng Song,Fengru Wang, Jerry Zhijian Yang
In this paper, we propose a method for solving semilinear elliptical equations using a ResNet with ${rm ReLU}^2$ activations. Firstly, we present a comprehensive formulation based on the penalized variational form of the elliptical equations. We then apply the Deep Ritz Method, which works for a wide range of equations. We obtain an upper bound on the errors between the acquired solutions and the true solutions in terms of the depth $mathcal{D},$ width $mathcal{W}$ of the ${rm ReLU}^2$ ResNet, and the number of training samples $n.$ Our simulation results demonstrate that our method can effectively overcome the curse of dimensionality and validate the theoretical results.
{"title":"Analysis of Deep Ritz Methods for Semilinear Elliptic Equations","authors":"Mo Chen,Yuling Jiao,Xiliang Lu,Pengcheng Song,Fengru Wang, Jerry Zhijian Yang","doi":"10.4208/nmtma.oa-2023-0058","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2023-0058","url":null,"abstract":"In this paper, we propose a method for solving semilinear elliptical equations using a ResNet with ${rm ReLU}^2$ activations. Firstly, we present a comprehensive\u0000formulation based on the penalized variational form of the elliptical equations. We\u0000then apply the Deep Ritz Method, which works for a wide range of equations. We\u0000obtain an upper bound on the errors between the acquired solutions and the true\u0000solutions in terms of the depth $mathcal{D},$ width $mathcal{W}$ of the ${rm ReLU}^2$ ResNet, and the number of training samples $n.$ Our simulation results demonstrate that our method can\u0000effectively overcome the curse of dimensionality and validate the theoretical results.","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":"51 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140003603","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}
Pub Date : 2023-10-01DOI: 10.4208/nmtma.oa-2023-0023
Raziyeh Erfanifar, Masoud Hajarian and Khosro Sayevand
{"title":"A Novel Iterative Method to Find the Moore-Penrose Inverse of a Tensor with Einstein Product","authors":"Raziyeh Erfanifar, Masoud Hajarian and Khosro Sayevand","doi":"10.4208/nmtma.oa-2023-0023","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2023-0023","url":null,"abstract":"","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135761888","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}
Pub Date : 2023-08-01DOI: 10.4208/nmtma.oa-2023-0007
Qi Hong, Yuezheng Gong and Jia Zhao
{"title":"A Physics-Informed Structure-Preserving Numerical Scheme for the Phase-Field Hydrodynamic Model of Ternary Fluid Flows","authors":"Qi Hong, Yuezheng Gong and Jia Zhao","doi":"10.4208/nmtma.oa-2023-0007","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2023-0007","url":null,"abstract":"","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49644574","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}
Pub Date : 2023-06-01DOI: 10.4208/nmtma.oa-2022-0143
Jing Rui
. In this work, we consider a combined finite element method for fully coupled nonlinear thermo-poroelastic model problems. The mixed finite element (MFE) method is used for the pressure, the characteristics finite element (CFE) method is used for the temperature, and the Galerkin finite element (GFE) method is used for the elastic displacement. The semi-discrete and fully discrete finite element schemes are established and the stability of this method is presented. We derive error estimates for the pressure, temperature and displacement. Several numerical examples are presented to confirm the accuracy of the method.
{"title":"The MFE-CFE-GFE Method for the Fully Coupled Quasi-Static Thermo-Poroelastic Problem","authors":"Jing Rui","doi":"10.4208/nmtma.oa-2022-0143","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2022-0143","url":null,"abstract":". In this work, we consider a combined finite element method for fully coupled nonlinear thermo-poroelastic model problems. The mixed finite element (MFE) method is used for the pressure, the characteristics finite element (CFE) method is used for the temperature, and the Galerkin finite element (GFE) method is used for the elastic displacement. The semi-discrete and fully discrete finite element schemes are established and the stability of this method is presented. We derive error estimates for the pressure, temperature and displacement. Several numerical examples are presented to confirm the accuracy of the method.","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44218724","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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