Lina Wang, Mingzhu Zhang, Hongjiong Tian, Lijun Yi
We introduce and analyze an $hp$-version $C^{1}$-continuous Petrov–Galerkin (CPG) method for nonlinear initial value problems of second-order ordinary differential equations. We derive a-priori error estimates in the $L^{2}$-, $L^{infty }$-, $H^{1}$- and $H^{2}$-norms that are completely explicit in the local time steps and local approximation degrees. Moreover, we show that the $hp$-version $C^{1}$-CPG method superconverges at the nodal points of the time partition with regard to the time steps and approximation degrees. As an application, we apply the $hp$-version $C^{1}$-CPG method to time discretization of nonlinear wave equations. Several numerical examples are presented to verify the theoretical results.
{"title":"hp-version C1-continuous Petrov–Galerkin method for nonlinear second-order initial value problems with application to wave equations","authors":"Lina Wang, Mingzhu Zhang, Hongjiong Tian, Lijun Yi","doi":"10.1093/imanum/drae036","DOIUrl":"https://doi.org/10.1093/imanum/drae036","url":null,"abstract":"We introduce and analyze an $hp$-version $C^{1}$-continuous Petrov–Galerkin (CPG) method for nonlinear initial value problems of second-order ordinary differential equations. We derive a-priori error estimates in the $L^{2}$-, $L^{infty }$-, $H^{1}$- and $H^{2}$-norms that are completely explicit in the local time steps and local approximation degrees. Moreover, we show that the $hp$-version $C^{1}$-CPG method superconverges at the nodal points of the time partition with regard to the time steps and approximation degrees. As an application, we apply the $hp$-version $C^{1}$-CPG method to time discretization of nonlinear wave equations. Several numerical examples are presented to verify the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141448749","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}
{"title":"Erratum to ‘Pressure robust SUPG-stabilized finite elements for the unsteady Navier–Stokes equation’","authors":"L. da Veiga, F. Dassi, G. Vacca","doi":"10.1093/imanum/drae035","DOIUrl":"https://doi.org/10.1093/imanum/drae035","url":null,"abstract":"","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141352628","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}
We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton–Jacobi–Bellman equations. We consider diffusion-corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations that are (formally) fractional order and second-order methods. It is well known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) strongly degenerate problems with Lipschitz solutions and (ii) weakly nondegenerate problems where we show that solutions have bounded fractional derivatives of order $sigma in (1,2)$. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly nondegenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than $mathcal{O}big (h^{frac{1}{2}}big )$.
{"title":"Precise error bounds for numerical approximations of fractional HJB equations","authors":"Indranil Chowdhury, Espen R Jakobsen","doi":"10.1093/imanum/drae030","DOIUrl":"https://doi.org/10.1093/imanum/drae030","url":null,"abstract":"We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton–Jacobi–Bellman equations. We consider diffusion-corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations that are (formally) fractional order and second-order methods. It is well known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) strongly degenerate problems with Lipschitz solutions and (ii) weakly nondegenerate problems where we show that solutions have bounded fractional derivatives of order $sigma in (1,2)$. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly nondegenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than $mathcal{O}big (h^{frac{1}{2}}big )$.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315615","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}
The Virtual Element Method (VEM) for diffusion-convection-reaction problems is considered. In order to design a quasi-robust scheme also in the convection-dominated regime, a Continuous Interior Penalty approach is employed. Due to the presence of polynomial projection operators, typical of the VEM, the stability and the error analysis requires particular care—especially in treating the advective term. Some numerical tests are presented to support the theoretical results.
{"title":"CIP-stabilized virtual elements for diffusion-convection-reaction problems","authors":"L Beirão da Veiga, C Lovadina, M Trezzi","doi":"10.1093/imanum/drae020","DOIUrl":"https://doi.org/10.1093/imanum/drae020","url":null,"abstract":"The Virtual Element Method (VEM) for diffusion-convection-reaction problems is considered. In order to design a quasi-robust scheme also in the convection-dominated regime, a Continuous Interior Penalty approach is employed. Due to the presence of polynomial projection operators, typical of the VEM, the stability and the error analysis requires particular care—especially in treating the advective term. Some numerical tests are presented to support the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141185236","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}
We investigate stochastic modified equations to explain the mathematical mechanism of symplectic methods applied to rough Hamiltonian systems. The contribution of this paper is threefold. First, we construct a new type of stochastic modified equation. For symplectic methods applied to rough Hamiltonian systems, the associated stochastic modified equations are proved to have Hamiltonian formulations. Secondly, the pathwise convergence order of the truncated modified equation to the numerical method is obtained by techniques in rough path theory. Thirdly, if increments of noises are simulated by truncated random variables, we show that the error can be made exponentially small with respect to the time step size.
{"title":"Stochastic modified equations for symplectic methods applied to rough Hamiltonian systems","authors":"Chuchu Chen, Jialin Hong, Chuying Huang","doi":"10.1093/imanum/drae019","DOIUrl":"https://doi.org/10.1093/imanum/drae019","url":null,"abstract":"We investigate stochastic modified equations to explain the mathematical mechanism of symplectic methods applied to rough Hamiltonian systems. The contribution of this paper is threefold. First, we construct a new type of stochastic modified equation. For symplectic methods applied to rough Hamiltonian systems, the associated stochastic modified equations are proved to have Hamiltonian formulations. Secondly, the pathwise convergence order of the truncated modified equation to the numerical method is obtained by techniques in rough path theory. Thirdly, if increments of noises are simulated by truncated random variables, we show that the error can be made exponentially small with respect to the time step size.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141096648","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}
We introduce a novel formulation for the evolution of parametric curves by anisotropic curve shortening flow in ${{mathbb{R}}}^{d}$, $dgeq 2$. The reformulation hinges on a suitable manipulation of the parameterization’s tangential velocity, leading to a strictly parabolic differential equation. Moreover, the derived equation is in divergence form, giving rise to a natural variational numerical method. For a fully discrete finite element approximation based on piecewise linear elements we prove optimal error estimates. Numerical simulations confirm the theoretical results and demonstrate the practicality of the method.
{"title":"Discrete anisotropic curve shortening flow in higher codimension","authors":"Klaus Deckelnick, Robert Nürnberg","doi":"10.1093/imanum/drae015","DOIUrl":"https://doi.org/10.1093/imanum/drae015","url":null,"abstract":"We introduce a novel formulation for the evolution of parametric curves by anisotropic curve shortening flow in ${{mathbb{R}}}^{d}$, $dgeq 2$. The reformulation hinges on a suitable manipulation of the parameterization’s tangential velocity, leading to a strictly parabolic differential equation. Moreover, the derived equation is in divergence form, giving rise to a natural variational numerical method. For a fully discrete finite element approximation based on piecewise linear elements we prove optimal error estimates. Numerical simulations confirm the theoretical results and demonstrate the practicality of the method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141096758","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}
We introduce in this paper the numerical analysis of high order both in time and space Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order $q=5$. The development and analysis of the methods are performed in the framework of time evolving finite elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical Analysis41, 1696–1845 (2021). The error estimates show through their dependence on the parameters of the equation the existence of different regimes in the behavior of the numerical solution; namely, in the diffusive regime, that is, when the diffusion parameter $mu $ is large, the error is $O(h^{k+1}+varDelta t^{q})$, whereas in the advective regime, $mu ll 1$, the convergence is $O(min (h^{k},frac{h^{k+1} }{varDelta t})+varDelta t^{q})$. It is worth remarking that the error constant does not have exponential $mu ^{-1}$ dependence.
本文介绍了针对平流-扩散方程保守公式的高阶时间和空间拉格朗日-加勒金方法的数值分析。作为时间离散化方案，我们考虑了最高阶数为 $q=5$ 的后向微分公式。方法的开发和分析是在 C. M. Elliot 和 T. Ranner, IMA Journal of Numerical Analysis41, 1696-1845 (2021) 中提出的时间演化有限元框架内进行的。误差估计值通过其对方程参数的依赖性显示出数值解的行为存在不同的体制；即在扩散状态下，即扩散参数 $mu $ 较大时，误差为 $O(h^{k+1}+varDelta t^{q})$ ；而在平流状态下，即 $mu ll 1$，收敛为 $O(min(h^{k},frac{h^{k+1}}{varDelta t})+varDelta t^{q})$ 。值得注意的是，误差常数与 $mu ^{-1}$不呈指数关系。
{"title":"High-order Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation","authors":"Rodolfo Bermejo, Manuel Colera","doi":"10.1093/imanum/drae018","DOIUrl":"https://doi.org/10.1093/imanum/drae018","url":null,"abstract":"We introduce in this paper the numerical analysis of high order both in time and space Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order $q=5$. The development and analysis of the methods are performed in the framework of time evolving finite elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical Analysis41, 1696–1845 (2021). The error estimates show through their dependence on the parameters of the equation the existence of different regimes in the behavior of the numerical solution; namely, in the diffusive regime, that is, when the diffusion parameter $mu $ is large, the error is $O(h^{k+1}+varDelta t^{q})$, whereas in the advective regime, $mu ll 1$, the convergence is $O(min (h^{k},frac{h^{k+1} }{varDelta t})+varDelta t^{q})$. It is worth remarking that the error constant does not have exponential $mu ^{-1}$ dependence.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140954632","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}
We propose a new a posteriori error estimator for mixed finite element discretizations of the curl-curl problem. This estimator relies on a Prager–Synge inequality, and therefore leads to fully guaranteed constant-free upper bounds on the error. The estimator is also locally efficient and polynomial-degree-robust. The construction is based on patch-wise divergence-constrained minimization problems, leading to a cheap embarrassingly parallel algorithm. Crucially, the estimator operates without any assumption on the topology of the domain, and unconventional arguments are required to establish the reliability estimate. Numerical examples illustrate the key theoretical results, and suggest that the estimator is suited for mesh adaptivity purposes.
{"title":"An equilibrated estimator for mixed finite element discretizations of the curl-curl problem","authors":"T Chaumont-Frelet","doi":"10.1093/imanum/drae007","DOIUrl":"https://doi.org/10.1093/imanum/drae007","url":null,"abstract":"We propose a new a posteriori error estimator for mixed finite element discretizations of the curl-curl problem. This estimator relies on a Prager–Synge inequality, and therefore leads to fully guaranteed constant-free upper bounds on the error. The estimator is also locally efficient and polynomial-degree-robust. The construction is based on patch-wise divergence-constrained minimization problems, leading to a cheap embarrassingly parallel algorithm. Crucially, the estimator operates without any assumption on the topology of the domain, and unconventional arguments are required to establish the reliability estimate. Numerical examples illustrate the key theoretical results, and suggest that the estimator is suited for mesh adaptivity purposes.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140954005","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}
In this paper, we present and analyze two linear and fully decoupled schemes for solving the unsteady incompressible magnetohydrodynamics equations. The rotational pressure-correction (RPC) approach is adopted to decouple the system, and the recently developed scalar auxiliary variable (SAV) method is used to treat the nonlinear terms explicitly and keep energy stability. One is the first-order RPC-SAV-Euler and the other one is generalized Crank–Nicolson-type scheme: GRPC-SAV-CN. For the RPC-SAV-Euler scheme, both unconditionally energy stability and optimal convergence are derived. The new GRPC-SAV-CN is constructed and can be regarded as a parameterized scheme, which includes PC-SAV-CN when the parameter $beta =0$ and RPC-SAV-CN when $beta in (0,frac {1}{2}]$; see Algorithm 3.2. However, Jiang and Yang (Jiang, N. & Yang, H. (2021) SIAM J. Sci. Comput., 43, A2869–A2896) point out that the SAV method has low accuracy by several commonly tested benchmark flow problem when solving Navier–Stokes equations. To improve the accuracy, we added two stabilization $-alpha _{1}varDelta tnu varDelta (widetilde {textbf {u}}^{n+1}-{textbf {u}}^{n})$ and $alpha _{2}varDelta tsigma ^{-1}mbox {curl}mbox {curl} (textbf {H}^{n+1}-textbf {H}^{n})$ in the GRPC-SAV-CN scheme, which play decisive roles in giving optimal error estimates. The unconditionally energy stability of the proposed scheme is given. We prove that the PC-SAV-CN scheme has second-order convergence speed, and the RPC-SAV-CN one has 1.5-order convergence rate. Finally, some numerical examples are presented to verify the validity and convergence of the numerical schemes.
{"title":"Optimal convergence analysis of two RPC-SAV schemes for the unsteady incompressible magnetohydrodynamics equations","authors":"Xiaojing Dong, Huayi Huang, Yunqing Huang, Xiaojuan Shen, Qili Tang","doi":"10.1093/imanum/drae016","DOIUrl":"https://doi.org/10.1093/imanum/drae016","url":null,"abstract":"In this paper, we present and analyze two linear and fully decoupled schemes for solving the unsteady incompressible magnetohydrodynamics equations. The rotational pressure-correction (RPC) approach is adopted to decouple the system, and the recently developed scalar auxiliary variable (SAV) method is used to treat the nonlinear terms explicitly and keep energy stability. One is the first-order RPC-SAV-Euler and the other one is generalized Crank–Nicolson-type scheme: GRPC-SAV-CN. For the RPC-SAV-Euler scheme, both unconditionally energy stability and optimal convergence are derived. The new GRPC-SAV-CN is constructed and can be regarded as a parameterized scheme, which includes PC-SAV-CN when the parameter $beta =0$ and RPC-SAV-CN when $beta in (0,frac {1}{2}]$; see Algorithm 3.2. However, Jiang and Yang (Jiang, N. & Yang, H. (2021) SIAM J. Sci. Comput., 43, A2869–A2896) point out that the SAV method has low accuracy by several commonly tested benchmark flow problem when solving Navier–Stokes equations. To improve the accuracy, we added two stabilization $-alpha _{1}varDelta tnu varDelta (widetilde {textbf {u}}^{n+1}-{textbf {u}}^{n})$ and $alpha _{2}varDelta tsigma ^{-1}mbox {curl}mbox {curl} (textbf {H}^{n+1}-textbf {H}^{n})$ in the GRPC-SAV-CN scheme, which play decisive roles in giving optimal error estimates. The unconditionally energy stability of the proposed scheme is given. We prove that the PC-SAV-CN scheme has second-order convergence speed, and the RPC-SAV-CN one has 1.5-order convergence rate. Finally, some numerical examples are presented to verify the validity and convergence of the numerical schemes.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140949377","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}
An enriched approximation space is the span of a conventional basis with a few extra functions included, for example to capture known features of the solution to a computational problem. Adding functions to a basis makes it overcomplete and, consequently, the corresponding discretized approximation problem may require solving an ill-conditioned system. Recent research indicates that these systems can still provide highly accurate numerical approximations under reasonable conditions. In this paper we propose an efficient algorithm to compute such approximations. It is based on the AZ algorithm for overcomplete sets and frames, which simplifies in the case of an enriched basis. In addition, analysis of the original AZ algorithm and of the proposed variant gives constructive insights on how to achieve optimal and stable discretizations using enriched bases. We apply the algorithm to examples of enriched approximation spaces in literature, including a few nonstandard approximation problems and an enriched spectral method for a 2D boundary value problem, and show that the simplified AZ algorithm is indeed stable, accurate and efficient.
丰富近似空间是在传统基础的跨度上加入一些额外的函数，例如为了捕捉计算问题解法的已知特征。在基上添加函数会使其过于完整，因此，相应的离散化近似问题可能需要求解一个条件不良的系统。最近的研究表明，在合理条件下，这些系统仍能提供高精度的数值近似。在本文中，我们提出了一种计算此类近似值的高效算法。该算法基于针对过完备集和框架的 AZ 算法，并在丰富基础的情况下进行了简化。此外，对原始 AZ 算法和所提变体的分析为如何利用丰富基实现最优和稳定离散提供了建设性见解。我们将该算法应用于文献中的富集近似空间实例，包括一些非标准近似问题和二维边界值问题的富集谱方法，结果表明简化的 AZ 算法确实稳定、精确且高效。
{"title":"Efficient function approximation in enriched approximation spaces","authors":"Astrid Herremans, Daan Huybrechs","doi":"10.1093/imanum/drae017","DOIUrl":"https://doi.org/10.1093/imanum/drae017","url":null,"abstract":"An enriched approximation space is the span of a conventional basis with a few extra functions included, for example to capture known features of the solution to a computational problem. Adding functions to a basis makes it overcomplete and, consequently, the corresponding discretized approximation problem may require solving an ill-conditioned system. Recent research indicates that these systems can still provide highly accurate numerical approximations under reasonable conditions. In this paper we propose an efficient algorithm to compute such approximations. It is based on the AZ algorithm for overcomplete sets and frames, which simplifies in the case of an enriched basis. In addition, analysis of the original AZ algorithm and of the proposed variant gives constructive insights on how to achieve optimal and stable discretizations using enriched bases. We apply the algorithm to examples of enriched approximation spaces in literature, including a few nonstandard approximation problems and an enriched spectral method for a 2D boundary value problem, and show that the simplified AZ algorithm is indeed stable, accurate and efficient.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140915068","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：