Abstract A finite element method with mass-lumping and flux upwinding is formulated for solving the immiscible two-phase flow problem in porous media. The method approximates directly thewetting phase pressure and saturation, which are the primary unknowns. The discrete saturation satisfies a maximum principle. Stability of the scheme and existence of a solution are established.
{"title":"A finite element method for degenerate two-phase flow in porous media. Part I: Well-posedness","authors":"V. Girault, B. Rivière, L. Cappanera","doi":"10.1515/JNMA-2020-0004","DOIUrl":"https://doi.org/10.1515/JNMA-2020-0004","url":null,"abstract":"Abstract A finite element method with mass-lumping and flux upwinding is formulated for solving the immiscible two-phase flow problem in porous media. The method approximates directly thewetting phase pressure and saturation, which are the primary unknowns. The discrete saturation satisfies a maximum principle. Stability of the scheme and existence of a solution are established.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2021-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82684439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Bounds on the spectrum of Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients of the convergence analysis of FETI (finite element tearing and interconnecting) based domain decomposition methods. Here we give bounds on the regular condition number of Schur complements of ‘floating’ clusters arising from the discretization of 3D Laplacian on a cube decomposed into cube subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m × m × m cube subdomains connected by face and optionally edge averages increases proportionally to m. The estimates support scalability of unpreconditioned H-FETI-DP (hybrid FETI dual-primal) method. Though the research is most important for the solution of variational inequalities, the results of numerical experiments indicate that unpreconditioned H-FETI-DP with large clusters can be useful also for the solution of huge linear problems.
子域刚度矩阵相对于内部变量的Schur补谱的界是基于FETI(有限元撕裂互连)域分解方法收敛性分析的关键因素。本文给出了三维拉普拉斯离散在分解成立方体子域的立方体上产生的“浮动”簇的Schur补的正则条件数的界。结果表明，在固定域上定义的聚类的条件数分解为m × m × m立方子域，这些子域由面和可选边平均连接，条件数随m成比例地增加。虽然该研究主要针对变分不等式的求解，但数值实验结果表明，具有大簇的无预条件H-FETI-DP也可用于求解大型线性问题。
{"title":"Schur complement spectral bounds for large hybrid FETI-DP clusters and huge three-dimensional scalar problems","authors":"Z. Dostál, T. Brzobohatý, O. Vlach","doi":"10.1515/JNMA-2020-0048","DOIUrl":"https://doi.org/10.1515/JNMA-2020-0048","url":null,"abstract":"Abstract Bounds on the spectrum of Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients of the convergence analysis of FETI (finite element tearing and interconnecting) based domain decomposition methods. Here we give bounds on the regular condition number of Schur complements of ‘floating’ clusters arising from the discretization of 3D Laplacian on a cube decomposed into cube subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m × m × m cube subdomains connected by face and optionally edge averages increases proportionally to m. The estimates support scalability of unpreconditioned H-FETI-DP (hybrid FETI dual-primal) method. Though the research is most important for the solution of variational inequalities, the results of numerical experiments indicate that unpreconditioned H-FETI-DP with large clusters can be useful also for the solution of huge linear problems.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2021-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77439560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We present simplified formulae for the analytic integration of the Newton potential of polynomials over boxes in two- and three-dimensional space. These are implemented in an easy-to-use C++ library that allows computations in arbitrary precision arithmetic which is also documented here. We describe how these results can be combined with fast multipole methods to evaluate the Newton potential of more general, non-polynomial densities.
{"title":"Analytic integration of the Newton potential over cuboids and an application to fast multipole methods","authors":"Matthias Kirchhart, Donat Weniger","doi":"10.1515/jnma-2020-0103","DOIUrl":"https://doi.org/10.1515/jnma-2020-0103","url":null,"abstract":"Abstract We present simplified formulae for the analytic integration of the Newton potential of polynomials over boxes in two- and three-dimensional space. These are implemented in an easy-to-use C++ library that allows computations in arbitrary precision arithmetic which is also documented here. We describe how these results can be combined with fast multipole methods to evaluate the Newton potential of more general, non-polynomial densities.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82839452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We explore algebraic strategies for numerically solving linear elliptic partial differential equations in polygonal domains. To discretize the polygon by means of structured meshes, we employ Schwarz–Christoffel conformal mappings, leading to a multiterm linear equation possibly including Hadamard products of some of the terms. This new algebraic formulation allows us to clearly distinguish between the role of the discretized operators and that of the domain meshing. Various algebraic strategies are discussed for the solution of the resulting matrix equation.
{"title":"Matrix equation solving of PDEs in polygonal domains using conformal mappings","authors":"Yuebin Hao, V. Simoncini","doi":"10.1515/jnma-2020-0035","DOIUrl":"https://doi.org/10.1515/jnma-2020-0035","url":null,"abstract":"Abstract We explore algebraic strategies for numerically solving linear elliptic partial differential equations in polygonal domains. To discretize the polygon by means of structured meshes, we employ Schwarz–Christoffel conformal mappings, leading to a multiterm linear equation possibly including Hadamard products of some of the terms. This new algebraic formulation allows us to clearly distinguish between the role of the discretized operators and that of the domain meshing. Various algebraic strategies are discussed for the solution of the resulting matrix equation.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76952323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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, we study the relative error in the numerical solution of a linear ordinary differential equation y'(t) = Ay(t), t ≥ 0, where A is a normal matrix. The numerical solution is obtained by using at any step an approximation of the matrix exponential, e.g., a polynomial or a rational approximation. The error of the numerical solution with respect to the exact solution is due to this approximation as well as to a possible perturbation in the initial value. For an unperturbed initial value, we have found: (1) unlike the absolute error, the relative error always grows linearly in time; (2) in the long-time, the contributions to the relative error relevant to non-rightmost eigenvalues of A disappear.
{"title":"Relative error analysis of matrix exponential approximations for numerical integration","authors":"S. Maset","doi":"10.1515/jnma-2020-0019","DOIUrl":"https://doi.org/10.1515/jnma-2020-0019","url":null,"abstract":"Abstract In this paper, we study the relative error in the numerical solution of a linear ordinary differential equation y'(t) = Ay(t), t ≥ 0, where A is a normal matrix. The numerical solution is obtained by using at any step an approximation of the matrix exponential, e.g., a polynomial or a rational approximation. The error of the numerical solution with respect to the exact solution is due to this approximation as well as to a possible perturbation in the initial value. For an unperturbed initial value, we have found: (1) unlike the absolute error, the relative error always grows linearly in time; (2) in the long-time, the contributions to the relative error relevant to non-rightmost eigenvalues of A disappear.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76111132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
F. Guillén-González, M. V. Redondo-Neble, J. Rodríguez-Galván
Abstract We propose a Discontinuous Galerkin (DG) scheme for the hydrostatic Stokes equations. These equations, related to large-scale PDE models in oceanography, are characterized by the loss of ellipticity of the vertical momentum equation. This fact provides some interesting challenges, such as the design of stable numerical schemes. The new scheme proposed here is based on the symmetric interior penalty (SIP) technique, with a particular treatment of the vertical velocity. It is well-known that stability of the mixed formulation of primitive equations requires, besides the LBB inf-sup condition, an additional hydrostatic inf-sup restriction relating pressure and vertical velocity. This hydrostatic inf-sup condition invalidates stability of usual Stokes stable continuous finite elements like Taylor-Hood 𝓟2/𝓟1 or bubble 𝓟1b/𝓟1. Here we prove stability for our 𝓟k/𝓟k DG scheme. Some novel numerical tests are provided which are in agreement with the previous analysis.
{"title":"Numerical analysis of a stable discontinuous Galerkin scheme for the hydrostatic Stokes problem","authors":"F. Guillén-González, M. V. Redondo-Neble, J. Rodríguez-Galván","doi":"10.1515/jnma-2019-0108","DOIUrl":"https://doi.org/10.1515/jnma-2019-0108","url":null,"abstract":"Abstract We propose a Discontinuous Galerkin (DG) scheme for the hydrostatic Stokes equations. These equations, related to large-scale PDE models in oceanography, are characterized by the loss of ellipticity of the vertical momentum equation. This fact provides some interesting challenges, such as the design of stable numerical schemes. The new scheme proposed here is based on the symmetric interior penalty (SIP) technique, with a particular treatment of the vertical velocity. It is well-known that stability of the mixed formulation of primitive equations requires, besides the LBB inf-sup condition, an additional hydrostatic inf-sup restriction relating pressure and vertical velocity. This hydrostatic inf-sup condition invalidates stability of usual Stokes stable continuous finite elements like Taylor-Hood 𝓟2/𝓟1 or bubble 𝓟1b/𝓟1. Here we prove stability for our 𝓟k/𝓟k DG scheme. Some novel numerical tests are provided which are in agreement with the previous analysis.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80706081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The methodology proposed in this paper bridges the gap between entropy stable and positivity-preserving discontinuous Galerkin (DG) methods for nonlinear hyperbolic problems. The entropy stability property and, optionally, preservation of local bounds for cell averages are enforced using flux limiters based on entropy conditions and discrete maximum principles, respectively. Entropy production by the (limited) gradients of the piecewise-linear DG approximation is constrained using Rusanov-type entropy viscosity. The Taylor basis representation of the entropy stabilization term reveals that it penalizes the solution gradients in a manner similar to slope limiting and requires implicit treatment to avoid severe time step restrictions. The optional application of a vertex-based slope limiter constrains the DG solution to be bounded by local maxima and minima of the cell averages. Numerical studies are performed for two scalar two-dimensional test problems with nonlinear and nonconvex flux functions.
{"title":"Entropy stabilization and property-preserving limiters for ℙ1 discontinuous Galerkin discretizations of scalar hyperbolic problems","authors":"D. Kuzmin","doi":"10.1515/jnma-2020-0056","DOIUrl":"https://doi.org/10.1515/jnma-2020-0056","url":null,"abstract":"Abstract The methodology proposed in this paper bridges the gap between entropy stable and positivity-preserving discontinuous Galerkin (DG) methods for nonlinear hyperbolic problems. The entropy stability property and, optionally, preservation of local bounds for cell averages are enforced using flux limiters based on entropy conditions and discrete maximum principles, respectively. Entropy production by the (limited) gradients of the piecewise-linear DG approximation is constrained using Rusanov-type entropy viscosity. The Taylor basis representation of the entropy stabilization term reveals that it penalizes the solution gradients in a manner similar to slope limiting and requires implicit treatment to avoid severe time step restrictions. The optional application of a vertex-based slope limiter constrains the DG solution to be bounded by local maxima and minima of the cell averages. Numerical studies are performed for two scalar two-dimensional test problems with nonlinear and nonconvex flux functions.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82732769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Fei Huang, Zuliang Lu, Lin Li, Xiankui Wu, Shang Liu, Yin Yang
Abstract With the climate change processes over times, all professions and trades in Three Gorges Reservoir Area will be influenced. One of the biggest challenges is the risk of rising sea level. In this situation, a large number of uncertainties for climate changes will be faced in Three Gorges Reservoir Area. Therefore, it is of importance to investigate the complexity of decision making on investing in the long term rising sea level risk related projects in Three Gorges Reservoir Area. This paper investigates the sea level and the temperature as the underlying assets in Three Gorges Reservoir Area. A real option model is constructed to evaluate potential sea level rising risk. We formulate European and American real option models into a linear parabolic variational inequalities and propose a power penalty approach to solve it. Then we obtain a nonlinear parabolic equation. It shows that the nonlinear parabolic equation is unique and solvable. Also, the solutions of the nonlinear parabolic equation converge to the solutions of the parabolic variational inequalities at the rate of order O(λ−k/2). Since the analytic solution of nonlinear parabolic equation is difficult to obtain, a fitted finite volume method is developed to solve it in case of European and American options, and the convergence of the nonlinear parabolic equation is obtained. An empirical analysis is presented to illustrate our theoretical results.
{"title":"Numerical simulation for European and American option of risks in climate change of Three Gorges Reservoir Area","authors":"Fei Huang, Zuliang Lu, Lin Li, Xiankui Wu, Shang Liu, Yin Yang","doi":"10.1515/jnma-2020-0081","DOIUrl":"https://doi.org/10.1515/jnma-2020-0081","url":null,"abstract":"Abstract With the climate change processes over times, all professions and trades in Three Gorges Reservoir Area will be influenced. One of the biggest challenges is the risk of rising sea level. In this situation, a large number of uncertainties for climate changes will be faced in Three Gorges Reservoir Area. Therefore, it is of importance to investigate the complexity of decision making on investing in the long term rising sea level risk related projects in Three Gorges Reservoir Area. This paper investigates the sea level and the temperature as the underlying assets in Three Gorges Reservoir Area. A real option model is constructed to evaluate potential sea level rising risk. We formulate European and American real option models into a linear parabolic variational inequalities and propose a power penalty approach to solve it. Then we obtain a nonlinear parabolic equation. It shows that the nonlinear parabolic equation is unique and solvable. Also, the solutions of the nonlinear parabolic equation converge to the solutions of the parabolic variational inequalities at the rate of order O(λ−k/2). Since the analytic solution of nonlinear parabolic equation is difficult to obtain, a fitted finite volume method is developed to solve it in case of European and American options, and the convergence of the nonlinear parabolic equation is obtained. An empirical analysis is presented to illustrate our theoretical results.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85623116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We introduce and analyze a stress-based formulation for Zener’s model in linear viscoelasticity. The method is aimed to tackle efficiently heterogeneous materials that admit purely elastic and viscoelastic parts in their composition.We write the mixed variational formulation of the problem in terms of a class of tensorial wave equation and obtain an energy estimate that guaranties the well-posedness of the problem through a standard Galerkin procedure. We propose and analyze mixed continuous and discontinuous Galerkin space discretizations of the problem and derive optimal error bounds for each semidiscrete solution in the corresponding energy norm. Finally, we discuss full discretization strategies for both Galerkin methods.
{"title":"Mixed-hybrid and mixed-discontinuous Galerkin methods for linear dynamical elastic–viscoelastic composite structures","authors":"A. M'arquez, S. Meddahi","doi":"10.1515/jnma-2020-0083","DOIUrl":"https://doi.org/10.1515/jnma-2020-0083","url":null,"abstract":"Abstract We introduce and analyze a stress-based formulation for Zener’s model in linear viscoelasticity. The method is aimed to tackle efficiently heterogeneous materials that admit purely elastic and viscoelastic parts in their composition.We write the mixed variational formulation of the problem in terms of a class of tensorial wave equation and obtain an energy estimate that guaranties the well-posedness of the problem through a standard Galerkin procedure. We propose and analyze mixed continuous and discontinuous Galerkin space discretizations of the problem and derive optimal error bounds for each semidiscrete solution in the corresponding energy norm. Finally, we discuss full discretization strategies for both Galerkin methods.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80204095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We introduce a new regularization model for incompressible fluid flow, which is a regularization of the EMAC (energy, momentum, and angular momentum conserving) formulation of the Navier–Stokes equations (NSE) that we call EMAC-Reg. The EMAC formulation has proved to be a useful formulation because it conserves energy, momentum, and angular momentum even when the divergence constraint is only weakly enforced. However, it is still a NSE formulation and so cannot resolve higher Reynolds number flows without very fine meshes. By carefully introducing regularization into the EMAC formulation, we create a model more suitable for coarser mesh computations but that still conserves the same quantities as EMAC, i.e., energy, momentum, and angular momentum. We show that EMAC-Reg, when semi-discretized with a finite element spatial discretization is well-posed and optimally accurate. Numerical results are provided that show EMAC-Reg is a robust coarse mesh model.
{"title":"An energy, momentum, and angular momentum conserving scheme for a regularization model of incompressible flow","authors":"Sean Ingimarson","doi":"10.1515/jnma-2020-0080","DOIUrl":"https://doi.org/10.1515/jnma-2020-0080","url":null,"abstract":"Abstract We introduce a new regularization model for incompressible fluid flow, which is a regularization of the EMAC (energy, momentum, and angular momentum conserving) formulation of the Navier–Stokes equations (NSE) that we call EMAC-Reg. The EMAC formulation has proved to be a useful formulation because it conserves energy, momentum, and angular momentum even when the divergence constraint is only weakly enforced. However, it is still a NSE formulation and so cannot resolve higher Reynolds number flows without very fine meshes. By carefully introducing regularization into the EMAC formulation, we create a model more suitable for coarser mesh computations but that still conserves the same quantities as EMAC, i.e., energy, momentum, and angular momentum. We show that EMAC-Reg, when semi-discretized with a finite element spatial discretization is well-posed and optimally accurate. Numerical results are provided that show EMAC-Reg is a robust coarse mesh model.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2020-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87075858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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