Aaron Brunk, Oliver Habrich, Timileyin David Oyedeji, Yangyiwei Yang, Bai-Xiang Xu
A Cahn–Hilliard–Allen–Cahn phase-field model coupled with a heat transfer equation, particularly with full non-diagonal mobility matrices, is studied. After reformulating the problem with respect to the inverse of temperature, we proposed and analysed a structure-preserving approximation for the semi-discretisation in space and then a fully discrete approximation using conforming finite elements and time-stepping methods. We prove structure-preserving property and discrete stability using relative entropy methods for the semi-discrete and fully discrete case. The theoretical results are illustrated by numerical experiments.
{"title":"Variational Approximation for a Non-Isothermal Coupled Phase-Field System: Structure-Preservation & Nonlinear Stability","authors":"Aaron Brunk, Oliver Habrich, Timileyin David Oyedeji, Yangyiwei Yang, Bai-Xiang Xu","doi":"10.1515/cmam-2023-0274","DOIUrl":"https://doi.org/10.1515/cmam-2023-0274","url":null,"abstract":"A Cahn–Hilliard–Allen–Cahn phase-field model coupled with a heat transfer equation, particularly with full non-diagonal mobility matrices, is studied. After reformulating the problem with respect to the inverse of temperature, we proposed and analysed a structure-preserving approximation for the semi-discretisation in space and then a fully discrete approximation using conforming finite elements and time-stepping methods. We prove structure-preserving property and discrete stability using relative entropy methods for the semi-discrete and fully discrete case. The theoretical results are illustrated by numerical experiments.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221716","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 formulate and analyze a space-time finite element method for the numerical simulation of rotating electric machines where the finite element mesh is fixed in a space-time domain. Based on the Babuška–Nečas theory we prove unique solvability both for the continuous variational formulation and for a standard Galerkin finite element discretization in the space-time domain. This approach allows for an adaptive resolution of the solution both in space and time, but it requires the solution of the overall system of algebraic equations. While the use of parallel solution algorithms seems to be mandatory, this also allows for a parallelization simultaneously in space and time. This approach is used for the eddy current approximation of the Maxwell equations which results in an elliptic-parabolic interface problem. Numerical results for linear and nonlinear constitutive material relations confirm the applicability and accuracy of the proposed approach.
{"title":"A Space-Time Finite Element Method for the Eddy Current Approximation of Rotating Electric Machines","authors":"Peter Gangl, Mario Gobrial, Olaf Steinbach","doi":"10.1515/cmam-2024-0033","DOIUrl":"https://doi.org/10.1515/cmam-2024-0033","url":null,"abstract":"In this paper we formulate and analyze a space-time finite element method for the numerical simulation of rotating electric machines where the finite element mesh is fixed in a space-time domain. Based on the Babuška–Nečas theory we prove unique solvability both for the continuous variational formulation and for a standard Galerkin finite element discretization in the space-time domain. This approach allows for an adaptive resolution of the solution both in space and time, but it requires the solution of the overall system of algebraic equations. While the use of parallel solution algorithms seems to be mandatory, this also allows for a parallelization simultaneously in space and time. This approach is used for the eddy current approximation of the Maxwell equations which results in an elliptic-parabolic interface problem. Numerical results for linear and nonlinear constitutive material relations confirm the applicability and accuracy of the proposed approach.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221722","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}
The free longitudinal vibrations of a rod are described by a differential equation of the form (P(x)y′)′+λP(x)y(x)=0{(P(x)yprime)^{prime}+lambda P(x)y(x)=0}, where P(x){P(x)} is the cross section area at point x and λ is an eigenvalue parameter. In this paper, first we discretize this differential equation by using the finite difference method to obtain a matrix eigenvalue problem of the form 𝐀Y=Λ𝐁Y{mathbf{A}Y=Lambdamathbf{B}Y}, where 𝐀{mathbf{A}} and 𝐁{mathbf{B}} are Jacobi and diagonal matrices dependent to cross section
杆的自由纵向振动由一个微分方程描述，其形式为 ( P ( x ) y ′ ) ′ + λ P ( x ) y ( x ) = 0 {(P(x)yprime)^{prime}+lambda P(x)y(x)=0} 。 其中，P ( x ) {P(x)} 是 x 点的横截面积，λ 是特征值参数。本文首先用有限差分法对该微分方程进行离散化，得到一个矩阵特征值问题，其形式为 𝐀 Y = Λ 𝐁 Y {mathbf{A}Y=Lambdamathbf{B}Y} 、其中，𝐀 {mathbf{A}} 和 𝐁 {mathbf{B}} 分别是与横截面 P ( x ) {P(x)} 相关的雅可比矩阵和对角矩阵。然后，我们通过修正所得矩阵特征值问题的特征值来估计杆方程的特征值。我们给出了一种基于修正思想的方法，通过求解逆矩阵特征值问题来构建横截面 P ( x ) {P(x)}。我们给出了一些数值示例来说明所提方法的效率。结果表明，该方法的收敛阶数为 O ( h 2 ) {O(h^{2})} 。
{"title":"An Inverse Matrix Eigenvalue Problem for Constructing a Vibrating Rod","authors":"Hanif Mirzaei, Vahid Abbasnavaz, Kazem Ghanbari","doi":"10.1515/cmam-2024-0001","DOIUrl":"https://doi.org/10.1515/cmam-2024-0001","url":null,"abstract":"The free longitudinal vibrations of a rod are described by a differential equation of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>P</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mi>y</m:mi> <m:mo>′</m:mo> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>′</m:mo> </m:msup> <m:mo>+</m:mo> <m:mi>λ</m:mi> <m:mi>P</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mi>y</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2024-0001_eq_0062.png\"/> <jats:tex-math>{(P(x)yprime)^{prime}+lambda P(x)y(x)=0}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>P</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2024-0001_eq_0397.png\"/> <jats:tex-math>{P(x)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the cross section area at point <jats:italic>x</jats:italic> and λ is an eigenvalue parameter. In this paper, first we discretize this differential equation by using the finite difference method to obtain a matrix eigenvalue problem of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>𝐀</m:mi> <m:mo></m:mo> <m:mi>Y</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi mathvariant=\"normal\">Λ</m:mi> <m:mo></m:mo> <m:mi>𝐁</m:mi> <m:mo></m:mo> <m:mi>Y</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2024-0001_eq_0437.png\"/> <jats:tex-math>{mathbf{A}Y=Lambdamathbf{B}Y}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝐀</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2024-0001_eq_0439.png\"/> <jats:tex-math>{mathbf{A}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝐁</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2024-0001_eq_0440.png\"/> <jats:tex-math>{mathbf{B}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> are Jacobi and diagonal matrices dependent to cross section <jats:inline-formula> <jats:alter","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141881211","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 propose and analyze a discontinuous Galerkin finite element method for solving the transient Boussinesq incompressible heat conducting fluid flow equations. This method utilizes an upwind approach to handle the nonlinear convective terms effectively. We discuss new a priori bounds for the semidiscrete discontinuous Galerkin approximations. Furthermore, we establish optimal a priori error estimates for the semidiscrete discontinuous Galerkin velocity approximation in L2mathbf{L}^{2} and energy norms, the temperature approximation in L2L^{2} and energy norms and pressure approximation in L2L^{2}-norm for t>0t>0. Additionally, under the smallness assumption on the data, we prove uniform in time error estimates. We also consider a backward Euler scheme for full discretization and derive fully discrete error estimates. Finally, we provide numerical examples to support the theoretical conclusions.
本文提出并分析了一种非连续 Galerkin 有限元方法，用于求解瞬态 Boussinesq 不可压缩导热流体流动方程。该方法利用上风法有效处理非线性对流项。我们讨论了半离散非连续 Galerkin 近似的新先验边界。此外，我们还为 L 2 mathbf{L}^{2} 和能量规范下的半离散不连续 Galerkin 速度近似、L 2 L^{2} 和能量规范下的温度近似以及 L 2 L^{2} -规范下的压力近似建立了最佳先验误差估计。 -t > 0 t>0 时的 L 2 L^{2} 和能量规范中的温度近似和 L 2 L^{2} 中的压力近似。此外，在数据较小的假设下，我们证明了时间误差估计的一致性。我们还考虑了完全离散化的后向欧拉方案，并推导出完全离散的误差估计值。最后，我们提供了数值示例来支持理论结论。
{"title":"On Error Estimates of a discontinuous Galerkin Method of the Boussinesq System of Equations","authors":"Saumya Bajpai, Debendra Kumar Swain","doi":"10.1515/cmam-2023-0202","DOIUrl":"https://doi.org/10.1515/cmam-2023-0202","url":null,"abstract":"In this paper, we propose and analyze a discontinuous Galerkin finite element method for solving the transient Boussinesq incompressible heat conducting fluid flow equations. This method utilizes an upwind approach to handle the nonlinear convective terms effectively. We discuss new a priori bounds for the semidiscrete discontinuous Galerkin approximations. Furthermore, we establish optimal a priori error estimates for the semidiscrete discontinuous Galerkin velocity approximation in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi mathvariant=\"bold\">L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0202_ineq_0001.png\"/> <jats:tex-math>mathbf{L}^{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and energy norms, the temperature approximation in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0202_ineq_0002.png\"/> <jats:tex-math>L^{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and energy norms and pressure approximation in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0202_ineq_0002.png\"/> <jats:tex-math>L^{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-norm for <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>t</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2023-0202_ineq_0004.png\"/> <jats:tex-math>t>0</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Additionally, under the smallness assumption on the data, we prove uniform in time error estimates. We also consider a backward Euler scheme for full discretization and derive fully discrete error estimates. Finally, we provide numerical examples to support the theoretical conclusions.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141587700","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 introduces the contents of the second of two special issues associated with the 9th International Conference on Computational Methods in Applied Mathematics, which took place from August 29 to September 2, 2022 in Vienna. It comments on the topics and highlights of all twelve papers of the special issue.
{"title":"Computational Methods in Applied Mathematics (CMAM 2022 Conference, Part 2)","authors":"Michael Feischl, Dirk Praetorius, Michele Ruggeri","doi":"10.1515/cmam-2024-0090","DOIUrl":"https://doi.org/10.1515/cmam-2024-0090","url":null,"abstract":"This paper introduces the contents of the second of two special issues associated with the 9th International Conference on Computational Methods in Applied Mathematics, which took place from August 29 to September 2, 2022 in Vienna. It comments on the topics and highlights of all twelve papers of the special issue.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508599","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}
The ability to deal with complex geometries and to go to higher orders is the main advantage of space-time finite element methods. Therefore, we want to develop a solid background from which we can construct appropriate space-time methods. In this paper, we will treat time as another space direction, which is the main idea of space-time methods. First, we will briefly discuss how exactly the vectorial wave equation is derived from Maxwell’s equations in a space-time structure, taking into account Ohm’s law. Then we will derive a space-time variational formulation for the vectorial wave equation using different trial and test spaces. This paper has two main goals. First, we prove unique solvability for the resulting Galerkin–Petrov variational formulation. Second, we analyze the discrete equivalent of the equation in a tensor product and show conditional stability, i.e., under a CFL condition. Understanding the vectorial wave equation and the corresponding space-time finite element methods is crucial for improving the existing theory of Maxwell’s equations and paves the way to computations of more complicated electromagnetic problems.
{"title":"Space-Time FEM for the Vectorial Wave Equation under Consideration of Ohm’s Law","authors":"Julia I. M. Hauser","doi":"10.1515/cmam-2023-0079","DOIUrl":"https://doi.org/10.1515/cmam-2023-0079","url":null,"abstract":"The ability to deal with complex geometries and to go to higher orders is the main advantage of space-time finite element methods. Therefore, we want to develop a solid background from which we can construct appropriate space-time methods. In this paper, we will treat time as another space direction, which is the main idea of space-time methods. First, we will briefly discuss how exactly the vectorial wave equation is derived from Maxwell’s equations in a space-time structure, taking into account Ohm’s law. Then we will derive a space-time variational formulation for the vectorial wave equation using different trial and test spaces. This paper has two main goals. First, we prove unique solvability for the resulting Galerkin–Petrov variational formulation. Second, we analyze the discrete equivalent of the equation in a tensor product and show conditional stability, i.e., under a CFL condition. Understanding the vectorial wave equation and the corresponding space-time finite element methods is crucial for improving the existing theory of Maxwell’s equations and paves the way to computations of more complicated electromagnetic problems.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141514360","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}
The p-Laplacian problem -∇⋅((μ+|∇u|p-2)∇u)=f{-nablacdot((mu+|nabla u|^{p-2})nabla u)=f} is considered, where μ is a given positive number. An anisotropic a posteriori residual-based error estimator is presented. The error estimator is shown to be equivalent, up to higher order terms, to the error in a quasi-norm. The involved constants being independent of μ, the solution, the mesh size and aspect ratio. An adaptive algorithm is proposed and numerical results are presented when p=3{p=3}. From this model problem, we propose a simplified error estimator and use it in the framework of an industrial application, namely a nonlinear Navier–Stokes problem arising from aluminium electrolysis.
考虑了 p-拉普拉斯问题-∇ ⋅ ( ( μ + | ∇ u | p - 2 ) ∇ u ) = f {-nablacdot((mu+|nabla u|^{p-2})nabla u)=f} ，其中 μ 是给定的正数。提出了一个基于各向异性后验残差的误差估计器。该误差估计器在高阶项上等同于准正则误差。所涉及的常数与 μ、解、网格尺寸和纵横比无关。我们提出了一种自适应算法，并给出了 p = 3 {p=3} 时的数值结果。根据这一模型问题，我们提出了一个简化的误差估计器，并将其用于工业应用框架，即铝电解产生的非线性纳维-斯托克斯问题。
{"title":"Anisotropic Adaptive Finite Elements for a p-Laplacian Problem","authors":"Paride Passelli, Marco Picasso","doi":"10.1515/cmam-2022-0205","DOIUrl":"https://doi.org/10.1515/cmam-2022-0205","url":null,"abstract":"The <jats:italic>p</jats:italic>-Laplacian problem <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⋅</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>μ</m:mi> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mi>f</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2022-0205_eq_0199.png\"/> <jats:tex-math>{-nablacdot((mu+|nabla u|^{p-2})nabla u)=f}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is considered, where μ is a given positive number. An anisotropic a posteriori residual-based error estimator is presented. The error estimator is shown to be equivalent, up to higher order terms, to the error in a quasi-norm. The involved constants being independent of μ, the solution, the mesh size and aspect ratio. An adaptive algorithm is proposed and numerical results are presented when <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2022-0205_eq_0387.png\"/> <jats:tex-math>{p=3}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. From this model problem, we propose a simplified error estimator and use it in the framework of an industrial application, namely a nonlinear Navier–Stokes problem arising from aluminium electrolysis.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508600","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, optimal convergence for an adaptive finite element algorithm for elastoplasticity is considered. To this end, the proposed adaptive algorithm is established within the abstract framework of the axioms of adaptivity [C. Carstensen, M. Feischl, M. Page and D. Praetorius, Axioms of adaptivity, Comput. Math. Appl. 67 2014, 6, 1195–1253], which provides a specific proceeding to prove the optimal convergence of the scheme. The proceeding is based on verifying four axioms, which ensure the optimal convergence. The verification is done by using results from [C. Carstensen, A. Schröder and S. Wiedemann, An optimal adaptive finite element method for elastoplasticity, Numer. Math. 132 2016, 1, 131–154], which presents an alternative approach to optimality without explicitly relying on the axioms
本文考虑了弹塑性自适应有限元算法的最佳收敛性。为此，本文在自适应公理的抽象框架内建立了所提出的自适应算法[C. Carstensen, M. Feischl, M. Page and D. Praetorius, 《弹性力学》]。Carstensen, M. Feischl, M. Page and D. Praetorius, Axioms of adaptivity, Comput.Math.67 2014, 6, 1195-1253]，它提供了证明方案最优收敛性的具体程序。该程序基于对四个公理的验证，这四个公理确保了最优收敛性。验证是利用 [C. Carstensen, A. Sch.Carstensen, A. Schröder and S. Wiedemann, An optimal adaptive finite element method for elastoplasticity, Numer.Math.132 2016, 1, 131-154] 中的结果，它提出了另一种不明确依赖公理的最优方法
{"title":"On an Optimal AFEM for Elastoplasticity","authors":"Miriam Schönauer, Andreas Schröder","doi":"10.1515/cmam-2024-0052","DOIUrl":"https://doi.org/10.1515/cmam-2024-0052","url":null,"abstract":"In this paper, optimal convergence for an adaptive finite element algorithm for elastoplasticity is considered. To this end, the proposed adaptive algorithm is established within the abstract framework of the axioms of adaptivity [C. Carstensen, M. Feischl, M. Page and D. Praetorius, Axioms of adaptivity, Comput. Math. Appl. 67 2014, 6, 1195–1253], which provides a specific proceeding to prove the optimal convergence of the scheme. The proceeding is based on verifying four axioms, which ensure the optimal convergence. The verification is done by using results from [C. Carstensen, A. Schröder and S. Wiedemann, An optimal adaptive finite element method for elastoplasticity, Numer. Math. 132 2016, 1, 131–154], which presents an alternative approach to optimality without explicitly relying on the axioms","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508601","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}
Abstract In this paper, a second-order algorithm based on the spectral deferred correction method is constructed for the time-dependent natural convection problem, which allows one to automatically increase the accuracy of a first-order backward-Euler time-stepping method through using spectral integration on Gaussian quadrature nodes and constructing the corrections. A complete theoretical analysis is presented to prove that this algorithm is unconditionally stable and possesses second-order accuracy in time. Numerical examples are given to confirm the theoretical analysis and the effectiveness of our algorithm.
{"title":"Numerical Analysis of a Second-Order Algorithm for the Time-Dependent Natural Convection Problem","authors":"Yiru Chen, Yun‐Bo Yang","doi":"10.1515/cmam-2023-0225","DOIUrl":"https://doi.org/10.1515/cmam-2023-0225","url":null,"abstract":"Abstract In this paper, a second-order algorithm based on the spectral deferred correction method is constructed for the time-dependent natural convection problem, which allows one to automatically increase the accuracy of a first-order backward-Euler time-stepping method through using spectral integration on Gaussian quadrature nodes and constructing the corrections. A complete theoretical analysis is presented to prove that this algorithm is unconditionally stable and possesses second-order accuracy in time. Numerical examples are given to confirm the theoretical analysis and the effectiveness of our algorithm.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141355766","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}
Daniele Boffi, Ondrej Certik, Francesca Gardini, Gianmarco Manzini
A Schrödinger equation for the system’s wavefunctions in a parallelepiped unit cell subject to Bloch-periodic boundary conditions must be solved repeatedly in quantum mechanical computations to derive the materials’ properties. Recent studies have demonstrated how enriched finite element type Galerkin methods can substantially lower the number of degrees of freedom necessary to produce accurate solutions with respect to the standard plane-waves method. In particular, the flat-top partition of unity finite element method enriched with the radial eigenfunctions of the one-dimensional Schrödinger equation offers a very effective way of solving the three-dimensional Schrödinger eigenvalue problem. We investigate the theoretical properties of this approximation method, its well-posedness and stability, we prove its convergence and derive suitable bound for the ℎ- and 𝑝-refinement in the L 2 L^{2} and energy norm for both the eigenvalues and the eigenfunctions. Finally, we confirm these theoretical results by applying this method to the eigenvalue problem of the one-electron Schrödinger equation with the harmonic potential, for which the exact solution is known.
在量子力学计算中，必须反复求解布洛赫周期边界条件下平行六面体单胞中系统波函数的薛定谔方程，才能得出材料的特性。最近的研究表明，与标准平面波方法相比，丰富的有限元类型 Galerkin 方法可以大大降低产生精确解法所需的自由度数量。特别是用一维薛定谔方程的径向特征函数丰富的平顶分割统一有限元方法，为解决三维薛定谔特征值问题提供了一种非常有效的方法。我们研究了这种近似方法的理论特性、拟合性和稳定性，证明了它的收敛性，并推导出了在 L 2 L^{2} 和能量规范下特征值和特征函数的𝑝-拟合的合适约束。最后，我们将这种方法应用于具有谐波势的单电子薛定谔方程的特征值问题，从而证实了这些理论结果。
{"title":"The Partition of Unity Finite Element Method for the Schrödinger Equation","authors":"Daniele Boffi, Ondrej Certik, Francesca Gardini, Gianmarco Manzini","doi":"10.1515/cmam-2023-0115","DOIUrl":"https://doi.org/10.1515/cmam-2023-0115","url":null,"abstract":"\u0000 A Schrödinger equation for the system’s wavefunctions in a parallelepiped unit cell subject to Bloch-periodic boundary conditions must be solved repeatedly in quantum mechanical computations to derive the materials’ properties.\u0000Recent studies have demonstrated how enriched finite element type Galerkin methods can substantially lower the number of degrees of freedom necessary to produce accurate solutions with respect to the standard plane-waves method.\u0000In particular, the flat-top partition of unity finite element method enriched with the radial eigenfunctions of the one-dimensional Schrödinger equation offers a very effective way of solving the three-dimensional Schrödinger eigenvalue problem.\u0000We investigate the theoretical properties of this approximation method, its well-posedness and stability, we prove its convergence and derive suitable bound for the ℎ- and 𝑝-refinement in the \u0000 \u0000 \u0000 \u0000 L\u0000 2\u0000 \u0000 \u0000 \u0000 L^{2}\u0000 \u0000 and energy norm for both the eigenvalues and the eigenfunctions.\u0000Finally, we confirm these theoretical results by applying this method to the eigenvalue problem of the one-electron Schrödinger equation with the harmonic potential, for which the exact solution is known.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140967548","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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