A discrete order-two Gagliardo–Nirenberg inequality is established for piecewise constant functions defined on a two-dimensional structured mesh composed of rectangular cells. As in the continuous framework, this discrete Gagliardo–Nirenberg inequality allows to control in particular the $L^4$ norm of the discrete gradient of the numerical solution by the $L^2$ norm of its discrete Hessian times its $L^infty $ norm. This result is crucial for the convergence analysis of a finite volume method for the approximation of a convection–diffusion equation involving a Joule effect term on a uniform mesh in each direction. The convergence proof relies on compactness arguments and on a priori estimates under a smallness assumption on the data, which is essential also in the continuous framework.
{"title":"Discrete Gagliardo–Nirenberg inequality and application to the finite volume approximation of a convection–diffusion equation with a Joule effect term","authors":"C. Calgaro, C. Cancès, E. Creusé","doi":"10.1093/imanum/drad063","DOIUrl":"https://doi.org/10.1093/imanum/drad063","url":null,"abstract":"A discrete order-two Gagliardo–Nirenberg inequality is established for piecewise constant functions defined on a two-dimensional structured mesh composed of rectangular cells. As in the continuous framework, this discrete Gagliardo–Nirenberg inequality allows to control in particular the $L^4$ norm of the discrete gradient of the numerical solution by the $L^2$ norm of its discrete Hessian times its $L^infty $ norm. This result is crucial for the convergence analysis of a finite volume method for the approximation of a convection–diffusion equation involving a Joule effect term on a uniform mesh in each direction. The convergence proof relies on compactness arguments and on a priori estimates under a smallness assumption on the data, which is essential also in the continuous framework.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48381868","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 work, following the discrete de Rham approach, we develop a discrete counterpart of a two-dimensional de Rham complex with enhanced regularity. The proposed construction supports general polygonal meshes and arbitrary approximation orders. We establish exactness on a contractible domain for both the versions of the complex with and without boundary conditions and, for the former, prove a complete set of Poincaré-type inequalities. The discrete complex is then used to derive a novel discretization method for a quad-rot problem, which, unlike other schemes in the literature, does not require the forcing term to be prepared. We carry out complete stability and convergence analyses for the proposed scheme and provide numerical validation of the results.
{"title":"An arbitrary-order discrete rot-rot complex on polygonal meshes with application to a quad-rot problem","authors":"Daniele Antonio Di Pietro","doi":"10.1093/imanum/drad045","DOIUrl":"https://doi.org/10.1093/imanum/drad045","url":null,"abstract":"Abstract In this work, following the discrete de Rham approach, we develop a discrete counterpart of a two-dimensional de Rham complex with enhanced regularity. The proposed construction supports general polygonal meshes and arbitrary approximation orders. We establish exactness on a contractible domain for both the versions of the complex with and without boundary conditions and, for the former, prove a complete set of Poincaré-type inequalities. The discrete complex is then used to derive a novel discretization method for a quad-rot problem, which, unlike other schemes in the literature, does not require the forcing term to be prepared. We carry out complete stability and convergence analyses for the proposed scheme and provide numerical validation of the results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136348995","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":"Correction to: On the stability of totally upwind schemes for the hyperbolic initial boundary value problem","authors":"","doi":"10.1093/imanum/drad068","DOIUrl":"https://doi.org/10.1093/imanum/drad068","url":null,"abstract":"","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48242424","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}
Gabriel R Barrenechea, Emmanuil H Georgoulis, Tristan Pryer, Andreas Veeser
Abstract This work proposes a nonlinear finite element method whose nodal values preserve bounds known for the exact solution. The discrete problem involves a nonlinear projection operator mapping arbitrary nodal values into bound-preserving ones and seeks the numerical solution in the range of this projection. As the projection is not injective, a stabilisation based upon the complementary projection is added in order to restore well-posedness. Within the framework of elliptic problems, the discrete problem may be viewed as a reformulation of a discrete obstacle problem, incorporating the inequality constraints through Lipschitz projections. The derivation of the proposed method is exemplified for linear and nonlinear reaction-diffusion problems. Near-best approximation results in suitable norms are established. In particular, we prove that, in the linear case, the numerical solution is the best approximation in the energy norm among all nodally bound-preserving finite element functions. A series of numerical experiments for such problems showcase the good behaviour of the proposed bound-preserving finite element method.
{"title":"A nodally bound-preserving finite element method","authors":"Gabriel R Barrenechea, Emmanuil H Georgoulis, Tristan Pryer, Andreas Veeser","doi":"10.1093/imanum/drad055","DOIUrl":"https://doi.org/10.1093/imanum/drad055","url":null,"abstract":"Abstract This work proposes a nonlinear finite element method whose nodal values preserve bounds known for the exact solution. The discrete problem involves a nonlinear projection operator mapping arbitrary nodal values into bound-preserving ones and seeks the numerical solution in the range of this projection. As the projection is not injective, a stabilisation based upon the complementary projection is added in order to restore well-posedness. Within the framework of elliptic problems, the discrete problem may be viewed as a reformulation of a discrete obstacle problem, incorporating the inequality constraints through Lipschitz projections. The derivation of the proposed method is exemplified for linear and nonlinear reaction-diffusion problems. Near-best approximation results in suitable norms are established. In particular, we prove that, in the linear case, the numerical solution is the best approximation in the energy norm among all nodally bound-preserving finite element functions. A series of numerical experiments for such problems showcase the good behaviour of the proposed bound-preserving finite element method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135236854","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 new high order approximations schemes for the Cox–Ingersoll–Ross (CIR) process that are obtained by using a recent technique developed by Alfonsi and Bally (2021, A generic construction for high order approximation schemes of semigroups using random grids. Numer. Math., 148, 743–793) for the approximation of semigroups. The idea consists in using a suitable combination of discretization schemes calculated on different random grids to increase the order of convergence. This technique coupled with the second order scheme proposed by Alfonsi (2010, High order discretization schemes for the CIR process: application to affine term structure and Heston models. Math. Comp., 79, 209–237) for the CIR leads to weak approximations of order $2k$, for all $kin{{mathbb{N}}}^{ast }$. Despite the singularity of the square-root volatility coefficient, we show rigorously this order of convergence under some restrictions on the volatility parameters. We illustrate numerically the convergence of these approximations for the CIR process and for the Heston stochastic volatility model and show the computational time gain they give.
{"title":"High order approximations of the Cox–Ingersoll–Ross process semigroup using random grids","authors":"Aurélien Alfonsi, Edoardo Lombardo","doi":"10.1093/imanum/drad059","DOIUrl":"https://doi.org/10.1093/imanum/drad059","url":null,"abstract":"Abstract We present new high order approximations schemes for the Cox–Ingersoll–Ross (CIR) process that are obtained by using a recent technique developed by Alfonsi and Bally (2021, A generic construction for high order approximation schemes of semigroups using random grids. Numer. Math., 148, 743–793) for the approximation of semigroups. The idea consists in using a suitable combination of discretization schemes calculated on different random grids to increase the order of convergence. This technique coupled with the second order scheme proposed by Alfonsi (2010, High order discretization schemes for the CIR process: application to affine term structure and Heston models. Math. Comp., 79, 209–237) for the CIR leads to weak approximations of order $2k$, for all $kin{{mathbb{N}}}^{ast }$. Despite the singularity of the square-root volatility coefficient, we show rigorously this order of convergence under some restrictions on the volatility parameters. We illustrate numerically the convergence of these approximations for the CIR process and for the Heston stochastic volatility model and show the computational time gain they give.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134983787","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 study a general class of singular degenerate parabolic stochastic partial differential equations (SPDEs) that include, in particular, the stochastic porous medium equations and the stochastic fast diffusion equation. We propose a fully discrete numerical approximation of the considered SPDEs based on the very weak formulation. By exploiting the monotonicity properties of the proposed formulation we prove the convergence of the numerical approximation towards the unique solution. Furthermore, we construct an implementable finite element scheme for the spatial discretization of the very weak formulation and provide numerical simulations to demonstrate the practicability of the proposed discretization.
{"title":"Numerical approximation of singular-degenerate parabolic stochastic partial differential equations","authors":"L. Baňas, B. Gess, C. Vieth","doi":"10.1093/imanum/drad061","DOIUrl":"https://doi.org/10.1093/imanum/drad061","url":null,"abstract":"\u0000 We study a general class of singular degenerate parabolic stochastic partial differential equations (SPDEs) that include, in particular, the stochastic porous medium equations and the stochastic fast diffusion equation. We propose a fully discrete numerical approximation of the considered SPDEs based on the very weak formulation. By exploiting the monotonicity properties of the proposed formulation we prove the convergence of the numerical approximation towards the unique solution. Furthermore, we construct an implementable finite element scheme for the spatial discretization of the very weak formulation and provide numerical simulations to demonstrate the practicability of the proposed discretization.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44342261","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 This paper introduces a novel approach for the construction of bulk–surface splitting schemes for semilinear parabolic partial differential equations with dynamic boundary conditions. The proposed construction is based on a reformulation of the system as a partial differential–algebraic equation and the inclusion of certain delay terms for the decoupling. To obtain a fully discrete scheme, the splitting approach is combined with finite elements in space and a backward differentiation formula in time. Within this paper, we focus on the second-order case, resulting in a $3$-step scheme. We prove second-order convergence under the assumption of a weak CFL-type condition and confirm the theoretical findings by numerical experiments. Moreover, we illustrate the potential for higher-order splitting schemes numerically.
{"title":"A second-order bulk–surface splitting for parabolic problems with dynamic boundary conditions","authors":"Robert Altmann, Christoph Zimmer","doi":"10.1093/imanum/drad062","DOIUrl":"https://doi.org/10.1093/imanum/drad062","url":null,"abstract":"Abstract This paper introduces a novel approach for the construction of bulk–surface splitting schemes for semilinear parabolic partial differential equations with dynamic boundary conditions. The proposed construction is based on a reformulation of the system as a partial differential–algebraic equation and the inclusion of certain delay terms for the decoupling. To obtain a fully discrete scheme, the splitting approach is combined with finite elements in space and a backward differentiation formula in time. Within this paper, we focus on the second-order case, resulting in a $3$-step scheme. We prove second-order convergence under the assumption of a weak CFL-type condition and confirm the theoretical findings by numerical experiments. Moreover, we illustrate the potential for higher-order splitting schemes numerically.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134977702","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 This paper derives optimized coefficients for optimized Schwarz iterations for the time-dependent Stokes–Darcy problem using an innovative strategy to solve a nonstandard min-max problem. The coefficients take into account both physical and discretization parameters that characterize the coupled problem, and they guarantee the robustness of the associated domain decomposition method. Numerical results validate the proposed approach in several test cases with physically relevant parameters.
{"title":"Optimized Schwarz methods for the time-dependent Stokes–Darcy coupling","authors":"Marco Discacciati, Tommaso Vanzan","doi":"10.1093/imanum/drad057","DOIUrl":"https://doi.org/10.1093/imanum/drad057","url":null,"abstract":"Abstract This paper derives optimized coefficients for optimized Schwarz iterations for the time-dependent Stokes–Darcy problem using an innovative strategy to solve a nonstandard min-max problem. The coefficients take into account both physical and discretization parameters that characterize the coupled problem, and they guarantee the robustness of the associated domain decomposition method. Numerical results validate the proposed approach in several test cases with physically relevant parameters.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136083409","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 work, we apply an a posteriori error analysis for the space-time, discontinuous in time, Galerkin scheme, which has been proposed in Antonopoulou (2020, Space-time discontinuous Galerkin methods for the $varepsilon $-dependent stochastic Allen–Cahn equation with mild noise. IMA J. Num. Analysis, 40, 2076–2105) for the $varepsilon $-dependent stochastic Allen–Cahn equation with mild noise $dot{W}^varepsilon $ tending to rough as $varepsilon rightarrow 0$. Our results are derived under low regularity since the noise even smooth in space is assumed only one-time continuously differentiable in time, according to the minimum regularity properties of Funaki (1999, Singular limit for stochastic reaction–diffusion equation and generation of random interfaces. Acta Math. Sinica, 15, 407–438). We prove a posteriori error estimates for the $m$-dimensional problem, $mleq 4$ for a general class of space-time finite element spaces. The a posteriori bound is growing only polynomially in $varepsilon ^{-1}$ if the step length $h$ is bounded by a positive power of $varepsilon $. This agrees with the restriction posed so far in the a priori error analysis of continuous finite element schemes for the $varepsilon $-dependent deterministic Allen–Cahn or deterministic and stochastic Cahn–Hilliard equation. As an application, we examine tensorial elements where the discrete solution is approximated by polynomial functions of separated space and time variables; the a posteriori estimates there involve dimensions, and the space, time discretization parameters. We then consider the special case of the mild noise $dot{W}^varepsilon $ as defined in Weber (2010, On the short time asymptotic of the stochastic Allen–Cahn equation. Ann. Inst. Henri Poincare Probab. Stat., 46, 965–975) through the convolution of a Gaussian process with a proper mollifying kernel, which is then numerically constructed. Finally, we provide some useful insights for the numerical algorithm, and present for the first time some numerical experiments of the scheme for both one- and two-dimensional problems in various cases of interest, and compare with the deterministic ones.
{"title":"A posteriori error analysis of space-time discontinuous Galerkin methods for the ε-stochastic Allen–Cahn equation","authors":"D. Antonopoulou, Bernard A. Egwu, Yubin Yan","doi":"10.1093/imanum/drad052","DOIUrl":"https://doi.org/10.1093/imanum/drad052","url":null,"abstract":"\u0000 In this work, we apply an a posteriori error analysis for the space-time, discontinuous in time, Galerkin scheme, which has been proposed in Antonopoulou (2020, Space-time discontinuous Galerkin methods for the $varepsilon $-dependent stochastic Allen–Cahn equation with mild noise. IMA J. Num. Analysis, 40, 2076–2105) for the $varepsilon $-dependent stochastic Allen–Cahn equation with mild noise $dot{W}^varepsilon $ tending to rough as $varepsilon rightarrow 0$. Our results are derived under low regularity since the noise even smooth in space is assumed only one-time continuously differentiable in time, according to the minimum regularity properties of Funaki (1999, Singular limit for stochastic reaction–diffusion equation and generation of random interfaces. Acta Math. Sinica, 15, 407–438). We prove a posteriori error estimates for the $m$-dimensional problem, $mleq 4$ for a general class of space-time finite element spaces. The a posteriori bound is growing only polynomially in $varepsilon ^{-1}$ if the step length $h$ is bounded by a positive power of $varepsilon $. This agrees with the restriction posed so far in the a priori error analysis of continuous finite element schemes for the $varepsilon $-dependent deterministic Allen–Cahn or deterministic and stochastic Cahn–Hilliard equation. As an application, we examine tensorial elements where the discrete solution is approximated by polynomial functions of separated space and time variables; the a posteriori estimates there involve dimensions, and the space, time discretization parameters. We then consider the special case of the mild noise $dot{W}^varepsilon $ as defined in Weber (2010, On the short time asymptotic of the stochastic Allen–Cahn equation. Ann. Inst. Henri Poincare Probab. Stat., 46, 965–975) through the convolution of a Gaussian process with a proper mollifying kernel, which is then numerically constructed. Finally, we provide some useful insights for the numerical algorithm, and present for the first time some numerical experiments of the scheme for both one- and two-dimensional problems in various cases of interest, and compare with the deterministic ones.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48726746","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 investigate the numerical approximation of integrals over $mathbb{R}^{d}$ equipped with the standard Gaussian measure $gamma $ for integrands belonging to the Gaussian-weighted Sobolev spaces $W^{alpha }_{p}(mathbb{R}^{d}, gamma )$ of mixed smoothness $alpha in mathbb{N}$ for $1 < p < infty $. We prove the asymptotic order of the convergence of optimal quadratures based on $n$ integration nodes and propose a novel method for constructing asymptotically optimal quadratures. As for related problems, we establish by a similar technique the asymptotic order of the linear, Kolmogorov and sampling $n$-widths in the Gaussian-weighted space $L_{q}(mathbb{R}^{d}, gamma )$ of the unit ball of $W^{alpha }_{p}(mathbb{R}^{d}, gamma )$ for $1 leq q < p < infty $ and $q=p=2$.
摘要本文研究了$mathbb{R}^{d}$上具有标准高斯测度$gamma $的混合光滑高斯加权Sobolev空间$W^{alpha }_{p}(mathbb{R}^{d}, gamma )$(对于$1 < p < infty $) $alpha in mathbb{N}$的积分的数值逼近。基于$n$积分节点证明了最优正交的渐近阶收敛性，提出了一种构造渐近最优正交的新方法。对于相关问题，我们用类似的方法建立了$1 leq q < p < infty $和$q=p=2$在单位球$W^{alpha }_{p}(mathbb{R}^{d}, gamma )$的高斯加权空间$L_{q}(mathbb{R}^{d}, gamma )$中线性宽度、Kolmogorov宽度和采样$n$ -宽度的渐近阶。
{"title":"Optimal numerical integration and approximation of functions on ℝ<i>d</i> equipped with Gaussian measure","authors":"Dinh Dũng, Van Kien Nguyen","doi":"10.1093/imanum/drad051","DOIUrl":"https://doi.org/10.1093/imanum/drad051","url":null,"abstract":"Abstract We investigate the numerical approximation of integrals over $mathbb{R}^{d}$ equipped with the standard Gaussian measure $gamma $ for integrands belonging to the Gaussian-weighted Sobolev spaces $W^{alpha }_{p}(mathbb{R}^{d}, gamma )$ of mixed smoothness $alpha in mathbb{N}$ for $1 &lt; p &lt; infty $. We prove the asymptotic order of the convergence of optimal quadratures based on $n$ integration nodes and propose a novel method for constructing asymptotically optimal quadratures. As for related problems, we establish by a similar technique the asymptotic order of the linear, Kolmogorov and sampling $n$-widths in the Gaussian-weighted space $L_{q}(mathbb{R}^{d}, gamma )$ of the unit ball of $W^{alpha }_{p}(mathbb{R}^{d}, gamma )$ for $1 leq q &lt; p &lt; infty $ and $q=p=2$.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135064454","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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