By postprocessing quadratic and eight‐node serendipity finite element solutions on arbitrary triangular and quadrilateral meshes, we obtain new quadratic/serendipity finite volume element solutions for solving anisotropic diffusion equations. The postprocessing procedure is implemented in each element independently, and we only need to solve a 6‐by‐6 (resp. 8‐by‐8) local linear algebraic system for triangular (resp. quadrilateral) element. The novelty of this paper is that, by designing some new quadratic dual meshes, and adding six/eight special constructed element‐wise bubble functions to quadratic/serendipity finite element solutions, we prove that the postprocessed solutions satisfy local conservation property on the new dual meshes. In particular, for any full anisotropic diffusion tensor, arbitrary triangular and quadrilateral meshes, we present a general framework to prove the existence and uniqueness of new quadratic/serendipity finite volume element solutions, which is better than some existing ones. That is, the existing theoretical results are improved, especially we extend the traditional rectangular assumption to arbitrary convex quadrilateral mesh. As a byproduct, we also prove that the new solutions converge to exact solution with optimal convergence rates under and norms on primal arbitrary triangular/quasi–parallelogram meshes. Finally, several numerical examples are carried out to validate the theoretical findings.
{"title":"New quadratic/serendipity finite volume element solutions on arbitrary triangular/quadrilateral meshes","authors":"Yanhui Zhou","doi":"10.1002/num.23093","DOIUrl":"https://doi.org/10.1002/num.23093","url":null,"abstract":"By postprocessing quadratic and eight‐node serendipity finite element solutions on arbitrary triangular and quadrilateral meshes, we obtain new quadratic/serendipity finite volume element solutions for solving anisotropic diffusion equations. The postprocessing procedure is implemented in each element independently, and we only need to solve a 6‐by‐6 (resp. 8‐by‐8) local linear algebraic system for triangular (resp. quadrilateral) element. The novelty of this paper is that, by designing some new quadratic dual meshes, and adding six/eight special constructed element‐wise bubble functions to quadratic/serendipity finite element solutions, we prove that the postprocessed solutions satisfy local conservation property on the new dual meshes. In particular, for any full anisotropic diffusion tensor, arbitrary triangular and quadrilateral meshes, we present a general framework to prove the existence and uniqueness of new quadratic/serendipity finite volume element solutions, which is better than some existing ones. That is, the existing theoretical results are improved, especially we extend the traditional rectangular assumption to arbitrary convex quadrilateral mesh. As a byproduct, we also prove that the new solutions converge to exact solution with optimal convergence rates under and norms on primal arbitrary triangular/quasi–parallelogram meshes. Finally, several numerical examples are carried out to validate the theoretical findings.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139836606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider Dirichlet problems of Laplace's equation in a bounded simply‐connected domain , and use the null field equation (NFE) of Green's representation formulation, where the source nodes are located on a pseudo‐boundary outside but not close to its boundary . Simple algorithms are proposed in this article by using the central rule for the NFE, and the normal derivatives of the solutions on the boundary can be easily obtained. These algorithms are called the discrete null field equation method (DNFEM) because the collocation equations are, indeed, the direct discrete form of the NFE. The bounds of the condition number are like those by the method of fundamental solutions (MFS) yielding the exponential growth as the number of unknowns increases. One trouble of the DNFEM is the near singularity of integrations for the solutions in boundary layers in Green's representation formulation. The traditional BEM also suffers from the same trouble. To deal with the near singularity, quadrature by expansions and the sinh transformation are often used. To handle this trouble, however, we develop two kinds of new techniques: (I) the interpolation techniques by Taylor's formulas with piecewise ‐degree polynomials and the Fourier series, and (II) the mini‐rules of integrals, such as the mini‐Simpson's and the mini‐Gaussian rules. Error analysis is made for technique I to achieve optimal convergence rates. Numerical experiments are carried out for disk domains to support the theoretical analysis made. The numerical performance of the DNFEM is excellent for disk domains to compete with the MFS. The errors with can be obtained by combination algorithms, which are satisfactory for most engineering problems. In summary, the new simple DNFEM is based on the NFE, which is different from the boundary element method (BEM). The theoretical basis in error and stability has been established in this article. One trouble in seeking the numerical solutions in boundary layers has been handled well; this is also an important contribution to the BEM. Besides, numerical experiments are encouraging. Hence the DNFEM is promising, and it may become a new boundary method for scientific/engineering computing.
考虑有界简单连接域中拉普拉斯方程的 Dirichlet 问题 ，并使用格林表征公式的空场方程 (NFE)，其中源节点位于其边界之外但不靠近其边界的伪边界上。本文利用 NFE 的中心法则提出了简单的算法，并可轻松求得边界上解的法导数。这些算法被称为离散空场方程法（DNFEM），因为配位方程实际上是 NFE 的直接离散形式。条件数的界限与基本解法（MFS）相似，随着未知数的增加而呈指数增长。DNFEM 的一个问题是，在格林表征公式中，边界层中的解的积分接近奇点。传统的 BEM 也存在同样的问题。为了解决近似奇异性问题，通常使用正交展开和 sinh 变换。然而，为了解决这一问题，我们开发了两种新技术：(I) 使用片断-度多项式和傅里叶级数的泰勒公式插值技术；(II) 积分的迷你规则，如迷你辛普森规则和迷你高斯规则。对技术 I 进行了误差分析，以达到最佳收敛率。为支持理论分析，对圆盘域进行了数值实验。DNFEM 在磁盘域的数值性能非常出色，可与 MFS 相媲美。通过组合算法可以获得与 MFS 的误差，这对于大多数工程问题来说都是令人满意的。总之，新的简单 DNFEM 基于 NFE，不同于边界元法（BEM）。本文建立了误差和稳定性的理论基础。在边界层中寻求数值解的一个难题得到了很好的解决；这也是对 BEM 的一个重要贡献。此外，数值实验也令人鼓舞。因此，DNFEM 前景广阔，有可能成为科学/工程计算的一种新边界方法。
{"title":"Discrete null field equation methods for solving Laplace's equation: Boundary layer computations","authors":"Li-Ping Zhang, Zi‐Cai Li, Ming-Gong Lee, Hung‐Tsai Huang","doi":"10.1002/num.23092","DOIUrl":"https://doi.org/10.1002/num.23092","url":null,"abstract":"Consider Dirichlet problems of Laplace's equation in a bounded simply‐connected domain , and use the null field equation (NFE) of Green's representation formulation, where the source nodes are located on a pseudo‐boundary outside but not close to its boundary . Simple algorithms are proposed in this article by using the central rule for the NFE, and the normal derivatives of the solutions on the boundary can be easily obtained. These algorithms are called the discrete null field equation method (DNFEM) because the collocation equations are, indeed, the direct discrete form of the NFE. The bounds of the condition number are like those by the method of fundamental solutions (MFS) yielding the exponential growth as the number of unknowns increases. One trouble of the DNFEM is the near singularity of integrations for the solutions in boundary layers in Green's representation formulation. The traditional BEM also suffers from the same trouble. To deal with the near singularity, quadrature by expansions and the sinh transformation are often used. To handle this trouble, however, we develop two kinds of new techniques: (I) the interpolation techniques by Taylor's formulas with piecewise ‐degree polynomials and the Fourier series, and (II) the mini‐rules of integrals, such as the mini‐Simpson's and the mini‐Gaussian rules. Error analysis is made for technique I to achieve optimal convergence rates. Numerical experiments are carried out for disk domains to support the theoretical analysis made. The numerical performance of the DNFEM is excellent for disk domains to compete with the MFS. The errors with can be obtained by combination algorithms, which are satisfactory for most engineering problems. In summary, the new simple DNFEM is based on the NFE, which is different from the boundary element method (BEM). The theoretical basis in error and stability has been established in this article. One trouble in seeking the numerical solutions in boundary layers has been handled well; this is also an important contribution to the BEM. Besides, numerical experiments are encouraging. Hence the DNFEM is promising, and it may become a new boundary method for scientific/engineering computing.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139777475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Numerical simulation of high‐speed flows often needs a fine grid for capturing detailed structures of shock or contact wave, which makes high‐order discontinuous Galerkin methods (DGMs) extremely costly. In this work, a characteristic‐compression based adaptive mesh refinement (AMR, h‐adaptive) method is proposed for efficiently improving resolution of the high‐speed flows. In order to allocate computational resources to needed regions, a characteristic‐compression embedded shock wave indicator is developed on incompatible grids and employed as the criterion for AMR. This indicator applies the admissible jumps of eigenvalues to measure the local compression of homogeneous characteristic curves, and theoretically can capture regions of characteristic‐compression which contain structures of shock, contact waves and vortices. Numerical results show that the proposed h‐adaptive DGM is robust, efficient and high‐resolution, it can capture dissipative shock, contact waves of different strengths and vortices with low noise on a rather coarse grid, and can significantly improve resolution of these structures through mild increase of computational resources as compared with the residual‐based h‐adaptive method.
高速流动的数值模拟通常需要精细网格来捕捉冲击波或接触波的细节结构，这使得高阶非连续伽勒金方法（DGM）的成本极高。本研究提出了一种基于特征压缩的自适应网格细化（AMR，h-adaptive）方法，以有效提高高速流动的分辨率。为了将计算资源分配到需要的区域，在不兼容网格上开发了一种嵌入式特征压缩冲击波指标，并将其作为 AMR 的标准。该指标利用特征值的可容许跃迁来测量均质特征曲线的局部压缩，理论上可以捕捉到包含冲击波、接触波和涡流结构的特征压缩区域。数值结果表明，与基于残差的 h 自适应方法相比，所提出的 h 自适应 DGM 具有鲁棒性、高效性和高分辨率的特点，它可以在相当粗的网格上捕获耗散冲击、不同强度的接触波和低噪声的涡流，并能通过轻微增加计算资源显著提高这些结构的分辨率。
{"title":"An adaptive mesh refinement method based on a characteristic‐compression embedded shock wave indicator for high‐speed flows","authors":"Yiwei Feng, Lili Lv, Tiegang Liu, Liang Xu, Weixiong Yuan","doi":"10.1002/num.23095","DOIUrl":"https://doi.org/10.1002/num.23095","url":null,"abstract":"Numerical simulation of high‐speed flows often needs a fine grid for capturing detailed structures of shock or contact wave, which makes high‐order discontinuous Galerkin methods (DGMs) extremely costly. In this work, a characteristic‐compression based adaptive mesh refinement (AMR, h‐adaptive) method is proposed for efficiently improving resolution of the high‐speed flows. In order to allocate computational resources to needed regions, a characteristic‐compression embedded shock wave indicator is developed on incompatible grids and employed as the criterion for AMR. This indicator applies the admissible jumps of eigenvalues to measure the local compression of homogeneous characteristic curves, and theoretically can capture regions of characteristic‐compression which contain structures of shock, contact waves and vortices. Numerical results show that the proposed h‐adaptive DGM is robust, efficient and high‐resolution, it can capture dissipative shock, contact waves of different strengths and vortices with low noise on a rather coarse grid, and can significantly improve resolution of these structures through mild increase of computational resources as compared with the residual‐based h‐adaptive method.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139839445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Numerical simulation of high‐speed flows often needs a fine grid for capturing detailed structures of shock or contact wave, which makes high‐order discontinuous Galerkin methods (DGMs) extremely costly. In this work, a characteristic‐compression based adaptive mesh refinement (AMR, h‐adaptive) method is proposed for efficiently improving resolution of the high‐speed flows. In order to allocate computational resources to needed regions, a characteristic‐compression embedded shock wave indicator is developed on incompatible grids and employed as the criterion for AMR. This indicator applies the admissible jumps of eigenvalues to measure the local compression of homogeneous characteristic curves, and theoretically can capture regions of characteristic‐compression which contain structures of shock, contact waves and vortices. Numerical results show that the proposed h‐adaptive DGM is robust, efficient and high‐resolution, it can capture dissipative shock, contact waves of different strengths and vortices with low noise on a rather coarse grid, and can significantly improve resolution of these structures through mild increase of computational resources as compared with the residual‐based h‐adaptive method.
高速流动的数值模拟通常需要精细网格来捕捉冲击波或接触波的细节结构，这使得高阶非连续伽勒金方法（DGM）的成本极高。本研究提出了一种基于特征压缩的自适应网格细化（AMR，h-adaptive）方法，以有效提高高速流动的分辨率。为了将计算资源分配到需要的区域，在不兼容网格上开发了一种嵌入式特征压缩冲击波指标，并将其作为 AMR 的标准。该指标利用特征值的可容许跃迁来测量均质特征曲线的局部压缩，理论上可以捕捉到包含冲击波、接触波和涡流结构的特征压缩区域。数值结果表明，与基于残差的 h 自适应方法相比，所提出的 h 自适应 DGM 具有鲁棒性、高效性和高分辨率的特点，它可以在相当粗的网格上捕获耗散冲击、不同强度的接触波和低噪声的涡流，并能通过轻微增加计算资源显著提高这些结构的分辨率。
{"title":"An adaptive mesh refinement method based on a characteristic‐compression embedded shock wave indicator for high‐speed flows","authors":"Yiwei Feng, Lili Lv, Tiegang Liu, Liang Xu, Weixiong Yuan","doi":"10.1002/num.23095","DOIUrl":"https://doi.org/10.1002/num.23095","url":null,"abstract":"Numerical simulation of high‐speed flows often needs a fine grid for capturing detailed structures of shock or contact wave, which makes high‐order discontinuous Galerkin methods (DGMs) extremely costly. In this work, a characteristic‐compression based adaptive mesh refinement (AMR, h‐adaptive) method is proposed for efficiently improving resolution of the high‐speed flows. In order to allocate computational resources to needed regions, a characteristic‐compression embedded shock wave indicator is developed on incompatible grids and employed as the criterion for AMR. This indicator applies the admissible jumps of eigenvalues to measure the local compression of homogeneous characteristic curves, and theoretically can capture regions of characteristic‐compression which contain structures of shock, contact waves and vortices. Numerical results show that the proposed h‐adaptive DGM is robust, efficient and high‐resolution, it can capture dissipative shock, contact waves of different strengths and vortices with low noise on a rather coarse grid, and can significantly improve resolution of these structures through mild increase of computational resources as compared with the residual‐based h‐adaptive method.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139779445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mario Álvarez, Eligio Colmenares, Filánder A. Sequeira
This paper presents our contribution to the a posteriori error analysis in 2D and 3D of a semi‐augmented mixed‐primal finite element method previously developed by us to numerically solve double‐diffusive natural convection problem in porous media. The model combines Brinkman‐Navier‐Stokes equations for velocity and pressure coupled to a vector advection‐diffusion equation, representing heat and concentration of a certain substance in a viscous fluid within a porous medium. Strain and pseudo‐stress tensors were introduced to establish scheme solvability and provide a priori error estimates using Raviart‐Thomas elements, piecewise polynomials and Lagrange finite elements. In this work, we derive two reliable residual‐based a posteriori error estimators. The first estimator leverages ellipticity properties, Helmholtz decomposition as well as Clément interpolant and Raviart‐Thomas operator properties for showing reliability; efficiency is guaranteed by inverse inequalities and localization strategies. An alternative estimator is also derived and analyzed for reliability without Helmholtz decomposition. Numerical tests are presented to confirm estimator properties and demonstrate adaptive scheme performance.
{"title":"A posteriori error analysis of a semi‐augmented finite element method for double‐diffusive natural convection in porous media","authors":"Mario Álvarez, Eligio Colmenares, Filánder A. Sequeira","doi":"10.1002/num.23090","DOIUrl":"https://doi.org/10.1002/num.23090","url":null,"abstract":"This paper presents our contribution to the a posteriori error analysis in 2D and 3D of a semi‐augmented mixed‐primal finite element method previously developed by us to numerically solve double‐diffusive natural convection problem in porous media. The model combines Brinkman‐Navier‐Stokes equations for velocity and pressure coupled to a vector advection‐diffusion equation, representing heat and concentration of a certain substance in a viscous fluid within a porous medium. Strain and pseudo‐stress tensors were introduced to establish scheme solvability and provide a priori error estimates using Raviart‐Thomas elements, piecewise polynomials and Lagrange finite elements. In this work, we derive two reliable residual‐based a posteriori error estimators. The first estimator leverages ellipticity properties, Helmholtz decomposition as well as Clément interpolant and Raviart‐Thomas operator properties for showing reliability; efficiency is guaranteed by inverse inequalities and localization strategies. An alternative estimator is also derived and analyzed for reliability without Helmholtz decomposition. Numerical tests are presented to confirm estimator properties and demonstrate adaptive scheme performance.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139839694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mario Álvarez, Eligio Colmenares, Filánder A. Sequeira
This paper presents our contribution to the a posteriori error analysis in 2D and 3D of a semi‐augmented mixed‐primal finite element method previously developed by us to numerically solve double‐diffusive natural convection problem in porous media. The model combines Brinkman‐Navier‐Stokes equations for velocity and pressure coupled to a vector advection‐diffusion equation, representing heat and concentration of a certain substance in a viscous fluid within a porous medium. Strain and pseudo‐stress tensors were introduced to establish scheme solvability and provide a priori error estimates using Raviart‐Thomas elements, piecewise polynomials and Lagrange finite elements. In this work, we derive two reliable residual‐based a posteriori error estimators. The first estimator leverages ellipticity properties, Helmholtz decomposition as well as Clément interpolant and Raviart‐Thomas operator properties for showing reliability; efficiency is guaranteed by inverse inequalities and localization strategies. An alternative estimator is also derived and analyzed for reliability without Helmholtz decomposition. Numerical tests are presented to confirm estimator properties and demonstrate adaptive scheme performance.
{"title":"A posteriori error analysis of a semi‐augmented finite element method for double‐diffusive natural convection in porous media","authors":"Mario Álvarez, Eligio Colmenares, Filánder A. Sequeira","doi":"10.1002/num.23090","DOIUrl":"https://doi.org/10.1002/num.23090","url":null,"abstract":"This paper presents our contribution to the a posteriori error analysis in 2D and 3D of a semi‐augmented mixed‐primal finite element method previously developed by us to numerically solve double‐diffusive natural convection problem in porous media. The model combines Brinkman‐Navier‐Stokes equations for velocity and pressure coupled to a vector advection‐diffusion equation, representing heat and concentration of a certain substance in a viscous fluid within a porous medium. Strain and pseudo‐stress tensors were introduced to establish scheme solvability and provide a priori error estimates using Raviart‐Thomas elements, piecewise polynomials and Lagrange finite elements. In this work, we derive two reliable residual‐based a posteriori error estimators. The first estimator leverages ellipticity properties, Helmholtz decomposition as well as Clément interpolant and Raviart‐Thomas operator properties for showing reliability; efficiency is guaranteed by inverse inequalities and localization strategies. An alternative estimator is also derived and analyzed for reliability without Helmholtz decomposition. Numerical tests are presented to confirm estimator properties and demonstrate adaptive scheme performance.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139779774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, a second-order time discretizing block-centered finite difference method is introduced to solve the compressible wormhole propagation. The optimal second-order error estimates for the porosity, pressure, velocity, concentration and its flux are established carefully in different discrete norms on non-uniform grids. Then by introducing Lagrange multiplier, a novel bound-preserving scheme for concentration is constructed. Finally, numerical experiments are carried out to demonstrate the correctness of theoretical analysis and capability for simulations of compressible wormhole propagation.
{"title":"A second-order time discretizing block-centered finite difference method for compressible wormhole propagation","authors":"Fei Sun, Xiaoli Li, Hongxing Rui","doi":"10.1002/num.23091","DOIUrl":"https://doi.org/10.1002/num.23091","url":null,"abstract":"In this paper, a second-order time discretizing block-centered finite difference method is introduced to solve the compressible wormhole propagation. The optimal second-order error estimates for the porosity, pressure, velocity, concentration and its flux are established carefully in different discrete norms on non-uniform grids. Then by introducing Lagrange multiplier, a novel bound-preserving scheme for concentration is constructed. Finally, numerical experiments are carried out to demonstrate the correctness of theoretical analysis and capability for simulations of compressible wormhole propagation.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139768163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Retraction: Nurgabyl DN, Uaissov AB. Asymptotic behavior of the solution of a singularly perturbed general boundary value problem with boundary jumps. Numer Methods Partial Differential Eq. 2021; 37: 2375–2392. https://doi.org/10.1002/num.22719
{"title":"Retraction: Asymptotic behavior of the solution of a singularly perturbed general boundary value problem with boundary jumps","authors":"","doi":"10.1002/num.23089","DOIUrl":"https://doi.org/10.1002/num.23089","url":null,"abstract":"<b>Retraction:</b> Nurgabyl DN, Uaissov AB. Asymptotic behavior of the solution of a singularly perturbed general boundary value problem with boundary jumps. <i>Numer Methods Partial Differential Eq</i>. 2021; 37: 2375–2392. https://doi.org/10.1002/num.22719","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop two totally decoupled, linear and second-order accurate numerical methods that are unconditionally energy stable for solving the Cahn–Hilliard–Darcy equations for two phase flows in porous media or in a Hele-Shaw cell. The implicit-explicit Crank–Nicolson leapfrog method is employed for the discretization of the Cahn–Hiliard equation to obtain linear schemes. Furthermore the artificial compression technique and pressure correction methods are utilized, respectively, so that the Cahn–Hiliard equation and the update of the Darcy pressure can be solved independently. We establish unconditionally long time stability of the schemes. Ample numerical experiments are performed to demonstrate the accuracy and robustness of the numerical methods, including simulations of the Rayleigh–Taylor instability, the Saffman–Taylor instability (fingering phenomenon).
{"title":"Fully decoupled unconditionally stable Crank–Nicolson leapfrog numerical methods for the Cahn–Hilliard–Darcy system","authors":"Yali Gao, Daozhi Han","doi":"10.1002/num.23087","DOIUrl":"https://doi.org/10.1002/num.23087","url":null,"abstract":"We develop two totally decoupled, linear and second-order accurate numerical methods that are unconditionally energy stable for solving the Cahn–Hilliard–Darcy equations for two phase flows in porous media or in a Hele-Shaw cell. The implicit-explicit Crank–Nicolson leapfrog method is employed for the discretization of the Cahn–Hiliard equation to obtain linear schemes. Furthermore the artificial compression technique and pressure correction methods are utilized, respectively, so that the Cahn–Hiliard equation and the update of the Darcy pressure can be solved independently. We establish unconditionally long time stability of the schemes. Ample numerical experiments are performed to demonstrate the accuracy and robustness of the numerical methods, including simulations of the Rayleigh–Taylor instability, the Saffman–Taylor instability (fingering phenomenon).","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139657304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the discrete velocity solution converges at the optimal order when solving the steady state Stokes equations by the - mixed finite element method for on 2D triangular grids or on tetrahedral grids, even in the case the inf-sup condition fails. By a simple -projection of the discrete pressure to the space of continuous polynomials, we show this post-processed pressure solution also converges at the optimal order. Both 2D and 3D numerical tests are presented, verifying the theory.
{"title":"On the convergence of 2D P4+ triangular and 3D P6+ tetrahedral divergence-free finite elements","authors":"Shangyou Zhang","doi":"10.1002/num.23088","DOIUrl":"https://doi.org/10.1002/num.23088","url":null,"abstract":"We show that the discrete velocity solution converges at the optimal order when solving the steady state Stokes equations by the <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0003\" display=\"inline\" location=\"graphic/num23088-math-0003.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msub>\u0000<mrow>\u0000<mi>P</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>k</mi>\u0000</mrow>\u0000</msub>\u0000</mrow>\u0000$$ {P}_k $$</annotation>\u0000</semantics></math>-<math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0004\" display=\"inline\" location=\"graphic/num23088-math-0004.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msubsup>\u0000<mrow>\u0000<mi>P</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>k</mi>\u0000<mo form=\"prefix\">−</mo>\u0000<mn>1</mn>\u0000</mrow>\u0000<mrow>\u0000<mtext>disc</mtext>\u0000</mrow>\u0000</msubsup>\u0000</mrow>\u0000$$ {P}_{k-1}^{mathrm{disc}} $$</annotation>\u0000</semantics></math> mixed finite element method for <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0005\" display=\"inline\" location=\"graphic/num23088-math-0005.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>k</mi>\u0000<mo>≥</mo>\u0000<mn>4</mn>\u0000</mrow>\u0000$$ kge 4 $$</annotation>\u0000</semantics></math> on 2D triangular grids or <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0006\" display=\"inline\" location=\"graphic/num23088-math-0006.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>k</mi>\u0000<mo>≥</mo>\u0000<mn>6</mn>\u0000</mrow>\u0000$$ kge 6 $$</annotation>\u0000</semantics></math> on tetrahedral grids, even in the case the inf-sup condition fails. By a simple <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0007\" display=\"inline\" location=\"graphic/num23088-math-0007.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msup>\u0000<mrow>\u0000<mi>L</mi>\u0000</mrow>\u0000<mrow>\u0000<mn>2</mn>\u0000</mrow>\u0000</msup>\u0000</mrow>\u0000$$ {L}^2 $$</annotation>\u0000</semantics></math>-projection of the discrete <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0008\" display=\"inline\" location=\"graphic/num23088-math-0008.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msub>\u0000<mrow>\u0000<mi>P</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>k</mi>\u0000<mo form=\"prefix\">−</mo>\u0000<mn>1</mn>\u0000</mrow>\u0000</msub>\u0000</mrow>\u0000$$ {P}_{k-1} $$</annotation>\u0000</semantics></math> pressure to the space of continuous <math altimg=\"urn:x-wiley:num:media:num23088:num23088-math-0009\" display=\"inline\" location=\"graphic/num23088-math-0009.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msub>\u0000<mrow>\u0000<mi>P</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>k</mi>\u0000<mo form=\"prefix\">−</mo>\u0000<mn>1</mn>\u0000</mrow>\u0000</msub>\u0000</mrow>\u0000$$ {P}_{k-1} $$</annotation>\u0000</semantics></math> polynomials, we show this post-processed pressure solution also converges at the optimal order. Both 2D and 3D numerical tests are presented, verifying the theory.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139560032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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