In this second part of our two-part article, we present and discuss the corresponding numerical results from implementations of the numerical algorithms described in the first part. With these results, we observed that • operator splitting applied to the associated time-dependent problem is suitable for solving only the first eigenproblem, • among those tried, the perturbation and arclength continuation approach was the sole effective and robust approach for solving higher eigenproblems, • on the eigenproblems for which (undamped or damped) Newton's method converged, it was without question the most efficient.
{"title":"On the numerical solution of a semilinear elliptic eigenproblem of Lane–Emden type, II: Numerical experiments","authors":"F. Foss, R. Glowinski, R. Hoppe","doi":"10.1515/jnum.2007.013","DOIUrl":"https://doi.org/10.1515/jnum.2007.013","url":null,"abstract":"In this second part of our two-part article, we present and discuss the corresponding numerical results from implementations of the numerical algorithms described in the first part. With these results, we observed that • operator splitting applied to the associated time-dependent problem is suitable for solving only the first eigenproblem, • among those tried, the perturbation and arclength continuation approach was the sole effective and robust approach for solving higher eigenproblems, • on the eigenproblems for which (undamped or damped) Newton's method converged, it was without question the most efficient.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128708544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We complete the analysis of our a posteriori error estimators for the time-dependent Stokes problem in Rd , d = 2 or 3. Our analysis covers non-conforming finite element approximation (Crouzeix–Raviart's elements) in space and backward Euler's scheme in time. For this discretization, we derived in part I of this paper [J. Numer. Math. (2007) 15, No. 2, 137–162] a residual indicator, which uses a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. In this second part we prove some analytical tools, and derive the lower and upper bounds of the spatial estimator.
我们完成了我们的后验误差估计的分析，时间相关的斯托克斯问题在Rd, d = 2或3。我们的分析涵盖了空间上的非协调有限元近似(Crouzeix-Raviart单元)和时间上的向后欧拉格式。对于这种离散化，我们在本文的第一部分中推导出[J]。号码。数学。[2007] [15]， No. 2, 137-162]残差指标，该指标使用基于非一致性近似的法向导数和切向导数跳跃的空间残差指标和基于每个时间步的破碎梯度跳跃的时间残差指标。在第二部分中，我们证明了一些分析工具，并推导了空间估计量的下界和上界。
{"title":"A posteriori error estimates for a nonconforming finite element discretization of the time-dependent Stokes problem, II: Analysis of the spatial estimator","authors":"S. Nicaise, N. Soualem","doi":"10.1515/jnma.2007.010","DOIUrl":"https://doi.org/10.1515/jnma.2007.010","url":null,"abstract":"We complete the analysis of our a posteriori error estimators for the time-dependent Stokes problem in Rd , d = 2 or 3. Our analysis covers non-conforming finite element approximation (Crouzeix–Raviart's elements) in space and backward Euler's scheme in time. For this discretization, we derived in part I of this paper [J. Numer. Math. (2007) 15, No. 2, 137–162] a residual indicator, which uses a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. In this second part we prove some analytical tools, and derive the lower and upper bounds of the spatial estimator.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125560759","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multiscale radiative heat transfer (RHT) problems are formulated and methods to approximate their numerical solutions are developed. We focus on RHT problems in participating media with heterogeneous optical properties leading to both optically thick and thin regimes within the same media spectrum. By introducing a diffusive scale and using an asymptotic expansion in the RHT equations we formulate the simplified PN approximations. The optical spectrum is decomposed in wavelength bands and the RHT equations are solved for bands with low absorption while the simplified PN equations are solved for bands with high absorption. The hybrid models solve the multiscale RHT more accurately than the simplified PN approximations and with a computational costs less than using the full RHT solver. Accuracy and effectiveness of the proposed models are demonstrated on three-dimensional RHT problems arising in combustion systems.
{"title":"A two-scale method for radiative heat transfer in non-grey absorbing and emitting media","authors":"Mohammed Seaïd","doi":"10.1515/jnma.2007.059","DOIUrl":"https://doi.org/10.1515/jnma.2007.059","url":null,"abstract":"Multiscale radiative heat transfer (RHT) problems are formulated and methods to approximate their numerical solutions are developed. We focus on RHT problems in participating media with heterogeneous optical properties leading to both optically thick and thin regimes within the same media spectrum. By introducing a diffusive scale and using an asymptotic expansion in the RHT equations we formulate the simplified PN approximations. The optical spectrum is decomposed in wavelength bands and the RHT equations are solved for bands with low absorption while the simplified PN equations are solved for bands with high absorption. The hybrid models solve the multiscale RHT more accurately than the simplified PN approximations and with a computational costs less than using the full RHT solver. Accuracy and effectiveness of the proposed models are demonstrated on three-dimensional RHT problems arising in combustion systems.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133106614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Existence statements for zeros of nonlinear equations are established via the interval Newton method generalizing previous theorems requiring regularity of the inclusion for the Jacobian.
利用区间牛顿法，推广了雅可比矩阵包含的正则性定理，建立了非线性方程零的存在性命题。
{"title":"Existence of a unique zero of nonlinear systems","authors":"P. Bao, F. Anton, J. Rokne","doi":"10.1515/jnma.2007.001","DOIUrl":"https://doi.org/10.1515/jnma.2007.001","url":null,"abstract":"Existence statements for zeros of nonlinear equations are established via the interval Newton method generalizing previous theorems requiring regularity of the inclusion for the Jacobian.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116967200","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Aeroacoustic simulations for low Machnumbers on the basis of the compressible NavierStokes equations result into a stiff multiscale problems, where the acoustic wave length and the wave length of the corresponding velocity perturbations are located on different scales and the speed of sound is larger by orders than the flow convection speed. Usually, aeroacoustic methods separates the multiple scales by solving the fluid flow without any acoustics, while the acoustic field is simulated afterwards or the stiffness is reduced by preconditioning techniques. An alternative approach, for which we restrict ourselves to smooth solutions, is presented here that solves the acoustic and the flow field fully coupled on an unstructured grid, which is designed taking into account the different length scales. However, the use of such an highly unstructured grid together with the stiffness of the problem, gives rise to a new numerical challenge: finding the optimal time step size for an equally distributed numerical error on the whole domain. The problem is solved using a fully implicit time discretization method. Due to the expected multiscale solution, the linear algebraic system of equations is solved with a geometric multigrid solver. It is possible to set up a multigrid procedure with Machnumber independent convergence rates, hence the solver is robust against the Machnumber.
{"title":"Low Machnumber aeroacoustics – A direct one-grid approach","authors":"A. Gordner, G. Wittum","doi":"10.1515/jnma.2007.007","DOIUrl":"https://doi.org/10.1515/jnma.2007.007","url":null,"abstract":"Aeroacoustic simulations for low Machnumbers on the basis of the compressible NavierStokes equations result into a stiff multiscale problems, where the acoustic wave length and the wave length of the corresponding velocity perturbations are located on different scales and the speed of sound is larger by orders than the flow convection speed. Usually, aeroacoustic methods separates the multiple scales by solving the fluid flow without any acoustics, while the acoustic field is simulated afterwards or the stiffness is reduced by preconditioning techniques. An alternative approach, for which we restrict ourselves to smooth solutions, is presented here that solves the acoustic and the flow field fully coupled on an unstructured grid, which is designed taking into account the different length scales. However, the use of such an highly unstructured grid together with the stiffness of the problem, gives rise to a new numerical challenge: finding the optimal time step size for an equally distributed numerical error on the whole domain. The problem is solved using a fully implicit time discretization method. Due to the expected multiscale solution, the linear algebraic system of equations is solved with a geometric multigrid solver. It is possible to set up a multigrid procedure with Machnumber independent convergence rates, hence the solver is robust against the Machnumber.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114901204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The degenerate isotropic boundary value problem –∇(ω2(x)∇u(x, y)) = f(x, y) on the unit square (0, 1)2 is considered in this paper. The weight function is assumed to be of the form ω2(ξ) = ξα, where α ≥ 0. This problem is discretized by piecewise linear finite elements on a triangular mesh of isosceles right triangles. The system of linear algebraic equations is solved by a preconditioned conjugate gradient method using a domain decomposition preconditioner with overlap. Two different preconditioners are presented and the optimality of the condition number for the preconditioned system is proved for α ≠ 1. The preconditioning operation requires O(N) operations, where N is the number of unknowns. Several numerical experiments show the performance of the proposed method.
{"title":"Overlapping additive Schwarz preconditioners for isotropic elliptic problems with degenerate coefficients","authors":"S. Beuchler, S. V. Nepomnyaschikh","doi":"10.1515/jnum.2007.012","DOIUrl":"https://doi.org/10.1515/jnum.2007.012","url":null,"abstract":"The degenerate isotropic boundary value problem –∇(ω2(x)∇u(x, y)) = f(x, y) on the unit square (0, 1)2 is considered in this paper. The weight function is assumed to be of the form ω2(ξ) = ξα, where α ≥ 0. This problem is discretized by piecewise linear finite elements on a triangular mesh of isosceles right triangles. The system of linear algebraic equations is solved by a preconditioned conjugate gradient method using a domain decomposition preconditioner with overlap. Two different preconditioners are presented and the optimality of the condition number for the preconditioned system is proved for α ≠ 1. The preconditioning operation requires O(N) operations, where N is the number of unknowns. Several numerical experiments show the performance of the proposed method.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115007127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper deals with various aspects of edge-oriented stabilization techniques for nonconforming finite element methods for the numerical solution of incompressible flow problems. We discuss two separate classes of problems which require appropriate stabilization techniques: First, the lack of coercivity for nonconforming low order approximations for treating problems with the symmetric deformation tensor instead of the gradient formulation in the momentum equation (‘Korn's inequality’) which particularly leads to convergence problems of the iterative solvers for small Reynolds (Re) numbers. Second, numerical instabilities for high Re numbers or whenever convective operators are dominant such that the standard Galerkin formulation fails and leads to spurious oscillations. We show that the right choice of edge-oriented stabilization is able to provide simultaneously excellent results regarding robustness and accuracy for both seemingly different cases of problems, and we discuss the sensitivity of the involved parameters w.r.t. variations of the Re number on unstructured meshes. Moreover, we explain how efficient multigrid solvers can be constructed to circumvent the problems with the arising ‘non-standard’ FEM data structures, and we provide several examples for the numerical efficiency for realistic flow configurations with benchmarking character.
{"title":"Unified edge-oriented stabilization of nonconforming FEM for incompressible flow problems: Numerical investigations","authors":"S. Turek, A. Ouazzi","doi":"10.1515/jnum.2007.014","DOIUrl":"https://doi.org/10.1515/jnum.2007.014","url":null,"abstract":"This paper deals with various aspects of edge-oriented stabilization techniques for nonconforming finite element methods for the numerical solution of incompressible flow problems. We discuss two separate classes of problems which require appropriate stabilization techniques: First, the lack of coercivity for nonconforming low order approximations for treating problems with the symmetric deformation tensor instead of the gradient formulation in the momentum equation (‘Korn's inequality’) which particularly leads to convergence problems of the iterative solvers for small Reynolds (Re) numbers. Second, numerical instabilities for high Re numbers or whenever convective operators are dominant such that the standard Galerkin formulation fails and leads to spurious oscillations. We show that the right choice of edge-oriented stabilization is able to provide simultaneously excellent results regarding robustness and accuracy for both seemingly different cases of problems, and we discuss the sensitivity of the involved parameters w.r.t. variations of the Re number on unstructured meshes. Moreover, we explain how efficient multigrid solvers can be constructed to circumvent the problems with the arising ‘non-standard’ FEM data structures, and we provide several examples for the numerical efficiency for realistic flow configurations with benchmarking character.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"88 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126198925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a convergence and stability analysis of the finite element modified method of characteristics for the incompressible Navier–Stokes equations. The method consists of combining a second-order backward time discretization based on the characteristics method with a spatial discretization of finite element type. We obtain stability results and optimal error estimates in the L 2-norm for velocity and pressure components under a time step restriction more relaxed than the standard Courant–Friedrichs–Levy condition. We also show some numerical results for two benchmark problems on the incompressible Navier–Stokes equations at different Reynolds numbers.
{"title":"Convergence and stability of finite element modified method of characteristics for the incompressible Navier–Stokes equations","authors":"Mofdi El-Amrani, Mohammed Seaïd","doi":"10.1515/jnma.2007.006","DOIUrl":"https://doi.org/10.1515/jnma.2007.006","url":null,"abstract":"We present a convergence and stability analysis of the finite element modified method of characteristics for the incompressible Navier–Stokes equations. The method consists of combining a second-order backward time discretization based on the characteristics method with a spatial discretization of finite element type. We obtain stability results and optimal error estimates in the L 2-norm for velocity and pressure components under a time step restriction more relaxed than the standard Courant–Friedrichs–Levy condition. We also show some numerical results for two benchmark problems on the incompressible Navier–Stokes equations at different Reynolds numbers.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132711874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider shape optimization of Stokes flow in channels where the objective is to design the lateral walls of the channel in such a way that a desired velocity profile is achieved. This amounts to the solution of a PDE constrained optimization problem with the state equation given by the Stokes system and the design variables being the control points of a Bézier curve representation of the lateral walls subject to bilateral constraints. Using a finite element discretization of the problem by Taylor–Hood elements, the shape optimization problem is solved numerically by a path-following primal-dual interior-point method applied to the parameter dependent nonlinear system representing the optimality conditions. The method is an all-at-once approach featuring an adaptive choice of the continuation parameter, inexact Newton solves by means of right-transforming iterations, and a monotonicity test for convergence monitoring. The performance of the adaptive continuation process is illustrated by several numerical examples.
{"title":"Path-following primal-dual interior-point methods for shape optimization of stationary flow problems","authors":"Harbir Antil, R. Hoppe, C. Linsenmann","doi":"10.1515/jnma.2007.005","DOIUrl":"https://doi.org/10.1515/jnma.2007.005","url":null,"abstract":"We consider shape optimization of Stokes flow in channels where the objective is to design the lateral walls of the channel in such a way that a desired velocity profile is achieved. This amounts to the solution of a PDE constrained optimization problem with the state equation given by the Stokes system and the design variables being the control points of a Bézier curve representation of the lateral walls subject to bilateral constraints. Using a finite element discretization of the problem by Taylor–Hood elements, the shape optimization problem is solved numerically by a path-following primal-dual interior-point method applied to the parameter dependent nonlinear system representing the optimality conditions. The method is an all-at-once approach featuring an adaptive choice of the continuation parameter, inexact Newton solves by means of right-transforming iterations, and a monotonicity test for convergence monitoring. The performance of the adaptive continuation process is illustrated by several numerical examples.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"104 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114068705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this first part of our two-part article, we present some theoretical background along with descriptions of some numerical techniques for solving a particular semilinear elliptic eigenproblem of Lane-Emden type on a triangular domain without any lines of symmetry. For solving the principal first eigenproblem, we describe an operator splitting method applied to the corresponding time-dependent problem. For solving higher eigenproblems, we describe an arclength continuation method applied to a particular perturbation of the original problem, which admits solution branches bifurcating from the trivial solution branch at eigenvalues of its linearization. We then solve the original eigenproblem by ‘jumping’ to a point on the unperturbed solution branch from a ‘nearby’ point on the corresponding continued perturbed branch, then normalizing the result. Finally, for comparison, we describe a particular implementation of Newton's method applied directly to the original constrained nonlinear eigenproblem.
{"title":"On the numerical solution of a semilinear elliptic eigenproblem of Lane–Emden type, I: Problem formulation and description of the algorithms","authors":"F. Foss, R. Glowinski, R. Hoppe","doi":"10.1515/jnma.2007.009","DOIUrl":"https://doi.org/10.1515/jnma.2007.009","url":null,"abstract":"In this first part of our two-part article, we present some theoretical background along with descriptions of some numerical techniques for solving a particular semilinear elliptic eigenproblem of Lane-Emden type on a triangular domain without any lines of symmetry. For solving the principal first eigenproblem, we describe an operator splitting method applied to the corresponding time-dependent problem. For solving higher eigenproblems, we describe an arclength continuation method applied to a particular perturbation of the original problem, which admits solution branches bifurcating from the trivial solution branch at eigenvalues of its linearization. We then solve the original eigenproblem by ‘jumping’ to a point on the unperturbed solution branch from a ‘nearby’ point on the corresponding continued perturbed branch, then normalizing the result. Finally, for comparison, we describe a particular implementation of Newton's method applied directly to the original constrained nonlinear eigenproblem.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2007-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132861842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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