In this paper, we investigate the nonlocal‐in‐time Allen‐Cahn equation (NiTACE), which incorporates a nonlocal operator in time with a finite nonlocal memory. Our objective is to examine the well‐posedness of the NiTACE by establishing the maximal regularity for the nonlocal‐in‐time parabolic equations with fractional power kernels. Furthermore, we derive a uniform energy bound by leveraging the positive definite property of kernel functions. We also develop an energy‐stable time stepping scheme specifically designed for the NiTACE. Additionally, we analyze the discrete maximum principle and energy dissipation law, which hold significant importance for phase field models. To ensure convergence, we verify the asymptotic compatibility of the proposed stable scheme. Lastly, we provide several numerical examples to illustrate the accuracy and effectiveness of our method.
{"title":"Analysis and numerical methods for nonlocal‐in‐time Allen‐Cahn equation","authors":"Hongwei Li, Jiang Yang, Wei Zhang","doi":"10.1002/num.23124","DOIUrl":"https://doi.org/10.1002/num.23124","url":null,"abstract":"In this paper, we investigate the nonlocal‐in‐time Allen‐Cahn equation (NiTACE), which incorporates a nonlocal operator in time with a finite nonlocal memory. Our objective is to examine the well‐posedness of the NiTACE by establishing the maximal regularity for the nonlocal‐in‐time parabolic equations with fractional power kernels. Furthermore, we derive a uniform energy bound by leveraging the positive definite property of kernel functions. We also develop an energy‐stable time stepping scheme specifically designed for the NiTACE. Additionally, we analyze the discrete maximum principle and energy dissipation law, which hold significant importance for phase field models. To ensure convergence, we verify the asymptotic compatibility of the proposed stable scheme. Lastly, we provide several numerical examples to illustrate the accuracy and effectiveness of our method.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141777543","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}
High order schemes are known to be unstable in the presence of shock discontinuities or under‐resolved solution features, and have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi‐discrete entropy inequality independently of discretization parameters. However, additional measures must be taken to ensure that solutions satisfy physical constraints such as positivity. In this work, we present a high order entropy stable discontinuous Galerkin (ESDG) method for the nonlinear shallow water equations (SWE) on two‐dimensional (2D) triangular meshes which preserves the positivity of the water heights. The scheme combines a low order positivity preserving method with a high order entropy stable method using convex limiting. This method is entropy stable and well‐balanced for fitted meshes with continuous bathymetry profiles.
{"title":"Entropy stable discontinuous Galerkin methods for the shallow water equations with subcell positivity preservation","authors":"Xinhui Wu, Nathaniel Trask, Jesse Chan","doi":"10.1002/num.23129","DOIUrl":"https://doi.org/10.1002/num.23129","url":null,"abstract":"High order schemes are known to be unstable in the presence of shock discontinuities or under‐resolved solution features, and have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi‐discrete entropy inequality independently of discretization parameters. However, additional measures must be taken to ensure that solutions satisfy physical constraints such as positivity. In this work, we present a high order entropy stable discontinuous Galerkin (ESDG) method for the nonlinear shallow water equations (SWE) on two‐dimensional (2D) triangular meshes which preserves the positivity of the water heights. The scheme combines a low order positivity preserving method with a high order entropy stable method using convex limiting. This method is entropy stable and well‐balanced for fitted meshes with continuous bathymetry profiles.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141777545","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 article, we develop stable and efficient numerical schemes for a binary fluid‐surfactant phase‐field model which consists of two Cahn–Hilliard type equations with respect to the free energy containing a Ginzburg–Landau double‐well potential, a logarithmic Flory–Huggins potential and a nonlinear coupling entropy. The numerical schemes, which are decoupled and linear, are established by the central difference spatial approximation in combination with the first‐ and second‐order exponential time differencing methods based on the convex splitting of the free energy. For the sake of the linearity of the schemes, the nonlinear terms, especially the logarithmic term, are approximated explicitly, which requires the bound preservation of the numerical solution to make the algorithm robust. We conduct the convergence analysis and prove the bound‐preserving property in details for both first‐ and second‐order schemes, where the high‐order consistency analysis is applied to the first‐order case. In addition, the energy stability is also obtained by the nature of the convex splitting. Numerical experiments are performed to verify the accuracy and stability of the schemes and simulate the dynamics of phase separation and surfactant adsorption.
{"title":"Convergence and stability analysis of energy stable and bound‐preserving numerical schemes for binary fluid‐surfactant phase‐field equations","authors":"Jiayi Duan, Xiao Li, Zhonghua Qiao","doi":"10.1002/num.23125","DOIUrl":"https://doi.org/10.1002/num.23125","url":null,"abstract":"In this article, we develop stable and efficient numerical schemes for a binary fluid‐surfactant phase‐field model which consists of two Cahn–Hilliard type equations with respect to the free energy containing a Ginzburg–Landau double‐well potential, a logarithmic Flory–Huggins potential and a nonlinear coupling entropy. The numerical schemes, which are decoupled and linear, are established by the central difference spatial approximation in combination with the first‐ and second‐order exponential time differencing methods based on the convex splitting of the free energy. For the sake of the linearity of the schemes, the nonlinear terms, especially the logarithmic term, are approximated explicitly, which requires the bound preservation of the numerical solution to make the algorithm robust. We conduct the convergence analysis and prove the bound‐preserving property in details for both first‐ and second‐order schemes, where the high‐order consistency analysis is applied to the first‐order case. In addition, the energy stability is also obtained by the nature of the convex splitting. Numerical experiments are performed to verify the accuracy and stability of the schemes and simulate the dynamics of phase separation and surfactant adsorption.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551400","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}
Yue Luo, Lei Zhang, Pingwen Zhang, Zhiyi Zhang, Xiangcheng Zheng
We analyze the semi‐implicit scheme of high‐index saddle dynamics, which provides a powerful numerical method for finding the any‐index saddle points and constructing the solution landscape. Compared with the explicit schemes of saddle dynamics, the semi‐implicit discretization relaxes the step size and accelerates the convergence, but the corresponding numerical analysis encounters new difficulties compared to the explicit scheme. Specifically, the orthonormal property of the eigenvectors at each time step could not be fully employed due to the semi‐implicit treatment, and computations of the eigenvectors are coupled with the orthonormalization procedure, which further complicates the numerical analysis. We address these issues to prove error estimates of the semi‐implicit scheme via, for example, technical splittings and multi‐variable circulating induction procedure. We further analyze the convergence rate of the generalized minimum residual solver for solving the semi‐implicit system. Extensive numerical experiments are carried out to substantiate the efficiency and accuracy of the semi‐implicit scheme in constructing solution landscapes of complex systems.
{"title":"Semi‐implicit method of high‐index saddle dynamics and application to construct solution landscape","authors":"Yue Luo, Lei Zhang, Pingwen Zhang, Zhiyi Zhang, Xiangcheng Zheng","doi":"10.1002/num.23123","DOIUrl":"https://doi.org/10.1002/num.23123","url":null,"abstract":"We analyze the semi‐implicit scheme of high‐index saddle dynamics, which provides a powerful numerical method for finding the any‐index saddle points and constructing the solution landscape. Compared with the explicit schemes of saddle dynamics, the semi‐implicit discretization relaxes the step size and accelerates the convergence, but the corresponding numerical analysis encounters new difficulties compared to the explicit scheme. Specifically, the orthonormal property of the eigenvectors at each time step could not be fully employed due to the semi‐implicit treatment, and computations of the eigenvectors are coupled with the orthonormalization procedure, which further complicates the numerical analysis. We address these issues to prove error estimates of the semi‐implicit scheme via, for example, technical splittings and multi‐variable circulating induction procedure. We further analyze the convergence rate of the generalized minimum residual solver for solving the semi‐implicit system. Extensive numerical experiments are carried out to substantiate the efficiency and accuracy of the semi‐implicit scheme in constructing solution landscapes of complex systems.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551401","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}
Jianfang Lu, Yan Jiang, Chi‐Wang Shu, Mengping Zhang
In this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results.
{"title":"Analysis of a class of spectral volume methods for linear scalar hyperbolic conservation laws","authors":"Jianfang Lu, Yan Jiang, Chi‐Wang Shu, Mengping Zhang","doi":"10.1002/num.23126","DOIUrl":"https://doi.org/10.1002/num.23126","url":null,"abstract":"In this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525514","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 article, we use the Nehari manifold and the eigenvalue problem for the negative Laplacian with Dirichlet boundary condition to analytically study the minimizers for the de Gennes–Cahn–Hilliard energy with quartic double‐well potential and Dirichlet boundary condition on the bounded domain. Our analysis reveals a bifurcation phenomenon determined by the boundary value and a bifurcation parameter that describes the thickness of the transition layer that segregates the binary mixture's two phases. Specifically, when the boundary value aligns precisely with the average of the pure phases, and the bifurcation parameter surpasses or equals a critical threshold, the minimizer assumes a unique form, representing the homogeneous state. Conversely, when the bifurcation parameter falls below this critical value, two symmetric minimizers emerge. Should the boundary value be larger or smaller from the average of the pure phases, symmetry breaks, resulting in a unique minimizer. Furthermore, we derive bounds of these minimizers, incorporating boundary conditions and features of the de Gennes–Cahn–Hilliard energy.
在本文中，我们利用 Nehari 流形和具有 Dirichlet 边界条件的负拉普拉奇的特征值问题，分析研究了有界域上具有四元双阱势和 Dirichlet 边界条件的 de Gennes-Cahn-Hilliard 能量的最小值。我们的分析揭示了一种由边界值和分岔参数决定的分岔现象，分岔参数描述了隔离二元混合物两相的过渡层的厚度。具体来说，当边界值与纯相的平均值精确一致，且分岔参数超过或等于临界阈值时，最小化器就会呈现唯一的形式，代表均相状态。反之，当分岔参数低于该临界值时，就会出现两个对称的最小化子。如果边界值大于或小于纯相的平均值，对称性就会被打破，从而产生一个独特的最小化器。此外，我们还结合边界条件和 de Gennes-Cahn-Hilliard 能量的特征，推导出了这些最小化器的边界。
{"title":"Minimizers for the de Gennes–Cahn–Hilliard energy under strong anchoring conditions","authors":"Shibin Dai, Abba Ramadan","doi":"10.1002/num.23127","DOIUrl":"https://doi.org/10.1002/num.23127","url":null,"abstract":"In this article, we use the Nehari manifold and the eigenvalue problem for the negative Laplacian with Dirichlet boundary condition to analytically study the minimizers for the de Gennes–Cahn–Hilliard energy with quartic double‐well potential and Dirichlet boundary condition on the bounded domain. Our analysis reveals a bifurcation phenomenon determined by the boundary value and a bifurcation parameter that describes the thickness of the transition layer that segregates the binary mixture's two phases. Specifically, when the boundary value aligns precisely with the average of the pure phases, and the bifurcation parameter surpasses or equals a critical threshold, the minimizer assumes a unique form, representing the homogeneous state. Conversely, when the bifurcation parameter falls below this critical value, two symmetric minimizers emerge. Should the boundary value be larger or smaller from the average of the pure phases, symmetry breaks, resulting in a unique minimizer. Furthermore, we derive bounds of these minimizers, incorporating boundary conditions and features of the de Gennes–Cahn–Hilliard energy.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525463","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}
K. Poochinapan, P. Manorot, T. Mouktonglang, B. Wongsaijai
Use of the finite difference method has produced successful solutions to the general partial differential equations due to its efficiency and effectiveness with wide applications. For example, the 2D Boussinesq paradigm equation can be numerically studied using a linear‐implicit finite difference scheme based on the Crank‐Nicolson/Adams‐Bashforth technique. First, conservative quantities are derived and preserved through numerical scheme. Then, the convergence and stability analysis is then provided to simulate a numerical solution whose existence and uniqueness are proved based on the boundedness of the numerical solution. Analysis of spatial accuracy is found to be second order on a uniform grid. Numerical results from simulations indicate that these proposed scheme provide satisfactory second‐order accuracy both in time and space with an ‐norm, and also preserve discrete invariants. Additionally, previous scientific literature review has provided little evidence of studied terms with dispersive effect in 2D Boussinesq paradigm equation. The current study explores solution behavior by applying the proposed scheme to numerically analyze initial Gaussian condition.
{"title":"A linear finite difference scheme with error analysis designed to preserve the structure of the 2D Boussinesq paradigm equation","authors":"K. Poochinapan, P. Manorot, T. Mouktonglang, B. Wongsaijai","doi":"10.1002/num.23119","DOIUrl":"https://doi.org/10.1002/num.23119","url":null,"abstract":"Use of the finite difference method has produced successful solutions to the general partial differential equations due to its efficiency and effectiveness with wide applications. For example, the 2D Boussinesq paradigm equation can be numerically studied using a linear‐implicit finite difference scheme based on the Crank‐Nicolson/Adams‐Bashforth technique. First, conservative quantities are derived and preserved through numerical scheme. Then, the convergence and stability analysis is then provided to simulate a numerical solution whose existence and uniqueness are proved based on the boundedness of the numerical solution. Analysis of spatial accuracy is found to be second order on a uniform grid. Numerical results from simulations indicate that these proposed scheme provide satisfactory second‐order accuracy both in time and space with an ‐norm, and also preserve discrete invariants. Additionally, previous scientific literature review has provided little evidence of studied terms with dispersive effect in 2D Boussinesq paradigm equation. The current study explores solution behavior by applying the proposed scheme to numerically analyze initial Gaussian condition.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531971","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}
Modeling the time evolution of atomic number densities and the kinetic (non‐Maxwellian) electron energy distribution function under the action of electron impact collisions by classical approaches requires an implicit time‐stepping scheme to maintain numerical stability. The resulting linear system that must be iteratively solved at each time step incorporates a dense (nonsparse) matrix. For variables being propagated, the computational cost is . We present an alternative approach with a computational cost of , which is the same order as the computational cost of an explicit method for propagating a system of this type. The approach relies on a combination of classical iterative derivative evaluations, combinatorial approximations, and some ideas from deep machine learning.
{"title":"Reducing the computational complexity of implicit schemes in the modeling of kinetic inelastic collisions in a partially ionized plasma","authors":"Carl Lederman, David Bilyeu","doi":"10.1002/num.23121","DOIUrl":"https://doi.org/10.1002/num.23121","url":null,"abstract":"Modeling the time evolution of atomic number densities and the kinetic (non‐Maxwellian) electron energy distribution function under the action of electron impact collisions by classical approaches requires an implicit time‐stepping scheme to maintain numerical stability. The resulting linear system that must be iteratively solved at each time step incorporates a dense (nonsparse) matrix. For variables being propagated, the computational cost is . We present an alternative approach with a computational cost of , which is the same order as the computational cost of an explicit method for propagating a system of this type. The approach relies on a combination of classical iterative derivative evaluations, combinatorial approximations, and some ideas from deep machine learning.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141342832","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}
Phase field models have been employed extensively in the study of microstructure evolution in materials. Elasticity plays an important role in solid‐state phase transformation processes, and it is usually introduced into phase field models in terms of the elastic strain energy by applying Khachaturyan–Shatalov microelasticity theory. Conventionally, this energy is derived in the reciprocal space and results in full‐space Fourier transformation in practice, which becomes bottle‐neck in large‐scale and massively‐parallel applications. In this article, we propose an error‐controlled approximation algorithm for scalable and efficient calculation of the elastic strain energy in phase field models. We first derive a real‐space convolutional representation of the elastic strain energy by representing the equilibrium displacements in the Khachaturyan–Shatalov microelasticity theory using Green's function. Then we propose an error‐controlled truncation criterion to approximate the corresponding terms in the phase field model. Finally, a carefully designed parallel algorithm is presented to carry out large‐scale simulations. The accuracy and efficiency of the proposed algorithm are demonstrated by real‐world large‐scale phase field simulations.
{"title":"A real space convolution‐based approximate algorithm for phase field model involving elastic strain energy","authors":"YaQian Gao, Xuebin Chi, JiXian Yin, Jian Zhang","doi":"10.1002/num.23122","DOIUrl":"https://doi.org/10.1002/num.23122","url":null,"abstract":"Phase field models have been employed extensively in the study of microstructure evolution in materials. Elasticity plays an important role in solid‐state phase transformation processes, and it is usually introduced into phase field models in terms of the elastic strain energy by applying Khachaturyan–Shatalov microelasticity theory. Conventionally, this energy is derived in the reciprocal space and results in full‐space Fourier transformation in practice, which becomes bottle‐neck in large‐scale and massively‐parallel applications. In this article, we propose an error‐controlled approximation algorithm for scalable and efficient calculation of the elastic strain energy in phase field models. We first derive a real‐space convolutional representation of the elastic strain energy by representing the equilibrium displacements in the Khachaturyan–Shatalov microelasticity theory using Green's function. Then we propose an error‐controlled truncation criterion to approximate the corresponding terms in the phase field model. Finally, a carefully designed parallel algorithm is presented to carry out large‐scale simulations. The accuracy and efficiency of the proposed algorithm are demonstrated by real‐world large‐scale phase field simulations.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141344261","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}
Rajan B. Adhikari, Imbumn Kim, Young Ju Lee, Dongwoo Sheen
We present an efficient numerical method to approximate the flux variable for the Darcy flow model. An important feature of our new method is that the approximate solution for the flux variable is obtained without approximating the pressure at all. To accomplish this, we introduce a user‐defined parameter delta, which is typically chosen to be small so that it minimizes the negative effect resulting from the absence of the pressure, such as inaccuracy in both the flux approximation and the mass conservation. The resulting algebraic system is of significantly smaller degrees of freedom, compared to the one from the mixed finite element methods or least‐squares methods. We also interpret the proposed method as a single step iterate of the augmented Lagrangian Uzawa applied to solve the mixed finite element in a special setting. Lastly, the pressure recovery from the flux variable is discussed and an optimal‐order error estimate for the method is obtained. Several examples are provided to verify the proposed theory and algorithm, some of which are from more realistic models such as SPE10.
{"title":"An efficient flux‐variable approximation scheme for Darcy's flow","authors":"Rajan B. Adhikari, Imbumn Kim, Young Ju Lee, Dongwoo Sheen","doi":"10.1002/num.23120","DOIUrl":"https://doi.org/10.1002/num.23120","url":null,"abstract":"We present an efficient numerical method to approximate the flux variable for the Darcy flow model. An important feature of our new method is that the approximate solution for the flux variable is obtained without approximating the pressure at all. To accomplish this, we introduce a user‐defined parameter delta, which is typically chosen to be small so that it minimizes the negative effect resulting from the absence of the pressure, such as inaccuracy in both the flux approximation and the mass conservation. The resulting algebraic system is of significantly smaller degrees of freedom, compared to the one from the mixed finite element methods or least‐squares methods. We also interpret the proposed method as a single step iterate of the augmented Lagrangian Uzawa applied to solve the mixed finite element in a special setting. Lastly, the pressure recovery from the flux variable is discussed and an optimal‐order error estimate for the method is obtained. Several examples are provided to verify the proposed theory and algorithm, some of which are from more realistic models such as SPE10.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141366237","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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