We prove two compactness results for function spaces with finite Dirichlet energy of half‐space nonlocal gradients. In each of these results, we provide sufficient conditions on a sequence of kernel functions that guarantee the asymptotic compact embedding of the associated nonlocal function spaces into the class of square‐integrable functions. Moreover, we will demonstrate that the sequence of nonlocal function spaces converges in an appropriate sense to a limiting function space. As an application, we prove uniform Poincaré‐type inequalities for sequence of half‐space gradient operators. We also apply the compactness result to demonstrate the convergence of appropriately parameterized nonlocal heterogeneous anisotropic diffusion problems. We will construct asymptotically compatible schemes for these type of problems. Another application concerns the convergence and robust discretization of a nonlocal optimal control problem.
{"title":"Compactness results for a Dirichlet energy of nonlocal gradient with applications","authors":"Zhaolong Han, Tadele Mengesha, Xiaochuan Tian","doi":"10.1002/num.23149","DOIUrl":"https://doi.org/10.1002/num.23149","url":null,"abstract":"We prove two compactness results for function spaces with finite Dirichlet energy of half‐space nonlocal gradients. In each of these results, we provide sufficient conditions on a sequence of kernel functions that guarantee the asymptotic compact embedding of the associated nonlocal function spaces into the class of square‐integrable functions. Moreover, we will demonstrate that the sequence of nonlocal function spaces converges in an appropriate sense to a limiting function space. As an application, we prove uniform Poincaré‐type inequalities for sequence of half‐space gradient operators. We also apply the compactness result to demonstrate the convergence of appropriately parameterized nonlocal heterogeneous anisotropic diffusion problems. We will construct asymptotically compatible schemes for these type of problems. Another application concerns the convergence and robust discretization of a nonlocal optimal control problem.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gradient‐based methods for training residual networks (ResNets) typically require a forward pass of input data, followed by back‐propagating the error gradient to update model parameters, which becomes time‐consuming as the network structure goes deeper. To break the algorithmic locking and exploit synchronous module parallelism in both forward and backward modes, auxiliary‐variable methods have emerged but suffer from communication overhead and a lack of data augmentation. By trading off the recomputation and storage of auxiliary variables, a joint learning framework is proposed in this work for training realistic ResNets across multiple compute devices. Specifically, the input data of each processor is generated from its low‐capacity auxiliary network (AuxNet), which permits the use of data augmentation and realizes forward unlocking. Backward passes are then executed in parallel, each with a local loss function derived from the penalty or augmented Lagrangian (AL) method. Finally, the AuxNet is adjusted to reproduce updated auxiliary variables through an end‐to‐end training process. We demonstrate the effectiveness of our method on ResNets and WideResNets across CIFAR‐10, CIFAR‐100, and ImageNet datasets, achieving speedup over the traditional layer‐serial training approach while maintaining comparable testing accuracy.
{"title":"Layer‐parallel training of residual networks with auxiliary variable networks","authors":"Qi Sun, Hexin Dong, Zewei Chen, Jiacheng Sun, Zhenguo Li, Bin Dong","doi":"10.1002/num.23147","DOIUrl":"https://doi.org/10.1002/num.23147","url":null,"abstract":"Gradient‐based methods for training residual networks (ResNets) typically require a forward pass of input data, followed by back‐propagating the error gradient to update model parameters, which becomes time‐consuming as the network structure goes deeper. To break the algorithmic locking and exploit synchronous module parallelism in both forward and backward modes, auxiliary‐variable methods have emerged but suffer from communication overhead and a lack of data augmentation. By trading off the recomputation and storage of auxiliary variables, a joint learning framework is proposed in this work for training realistic ResNets across multiple compute devices. Specifically, the input data of each processor is generated from its low‐capacity auxiliary network (AuxNet), which permits the use of data augmentation and realizes forward unlocking. Backward passes are then executed in parallel, each with a local loss function derived from the penalty or augmented Lagrangian (AL) method. Finally, the AuxNet is adjusted to reproduce updated auxiliary variables through an end‐to‐end training process. We demonstrate the effectiveness of our method on ResNets and WideResNets across CIFAR‐10, CIFAR‐100, and ImageNet datasets, achieving speedup over the traditional layer‐serial training approach while maintaining comparable testing accuracy.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The multilevel fast multipole method (MLFMM) is widely used to accelerate the solutions of acoustic and electromagnetic scattering problems. In the expansions and translation operators of the MLFMM for 3‐D scattering problems, some special functions are used, including spherical Bessel functions, spherical harmonics and Wigner symbol. This makes it difficult to analyze the truncation errors. In this paper, we first give sharp bounds for the truncation errors of the expansions used in the MLFMM, then derive the overall error formula of the MLFMM and estimate its upper bound, the result is finally applied to the cube octree structure. Some numerical examples are performed to validate the proposed results. The method in this paper can also be used to the MLFMM for other 3‐D problems, such as potential problems, elastostatic problems, Stokes flow problems and so on.
{"title":"Error bound of the multilevel fast multipole method for 3‐D scattering problems","authors":"Wenhui Meng","doi":"10.1002/num.23148","DOIUrl":"https://doi.org/10.1002/num.23148","url":null,"abstract":"The multilevel fast multipole method (MLFMM) is widely used to accelerate the solutions of acoustic and electromagnetic scattering problems. In the expansions and translation operators of the MLFMM for 3‐D scattering problems, some special functions are used, including spherical Bessel functions, spherical harmonics and Wigner symbol. This makes it difficult to analyze the truncation errors. In this paper, we first give sharp bounds for the truncation errors of the expansions used in the MLFMM, then derive the overall error formula of the MLFMM and estimate its upper bound, the result is finally applied to the cube octree structure. Some numerical examples are performed to validate the proposed results. The method in this paper can also be used to the MLFMM for other 3‐D problems, such as potential problems, elastostatic problems, Stokes flow problems and so on.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we present a formally fourth‐order accurate hybrid‐variable (HV) method for the Euler equations in the context of method of lines. The HV method seeks numerical approximations to both cell averages and nodal solutions and evolves them in time simultaneously; and it is proved in previous work that these methods are supraconvergent, that is, the order of the method is higher than that of the local truncation error. Taking advantage of the supraconvergence, the method is built on a third‐order discrete differential operator, which approximates the first spatial derivative at each grid point using only the information in the two neighboring cells. Analyses of stability, accuracy, and pointwise convergence are conducted in the one‐dimensional case for the linear advection equation; whereas extension to nonlinear systems including the Euler equations is achieved using characteristic decomposition and the incorporation of a residual‐consistent viscosity to capture strong discontinuities. Extensive numerical tests are presented to assess the numerical performance of the method for both 1D and 2D problems.
{"title":"An explicit fourth‐order hybrid‐variable method for Euler equations with a residual‐consistent viscosity","authors":"Xianyi Zeng","doi":"10.1002/num.23146","DOIUrl":"https://doi.org/10.1002/num.23146","url":null,"abstract":"In this article, we present a formally fourth‐order accurate hybrid‐variable (HV) method for the Euler equations in the context of method of lines. The HV method seeks numerical approximations to both cell averages and nodal solutions and evolves them in time simultaneously; and it is proved in previous work that these methods are supraconvergent, that is, the order of the method is higher than that of the local truncation error. Taking advantage of the supraconvergence, the method is built on a third‐order discrete differential operator, which approximates the first spatial derivative at each grid point using only the information in the two neighboring cells. Analyses of stability, accuracy, and pointwise convergence are conducted in the one‐dimensional case for the linear advection equation; whereas extension to nonlinear systems including the Euler equations is achieved using characteristic decomposition and the incorporation of a residual‐consistent viscosity to capture strong discontinuities. Extensive numerical tests are presented to assess the numerical performance of the method for both 1D and 2D problems.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, an efficient and stable exponential time difference method is presented for solving boundary layer problems. By combining exponential time difference schemes with spatial direct discontinuous Galerkin discretization based on exponential boundary layer approximations, the proposed algorithm not only may admit large time step sizes but also could provide good spatial approximations even if on rather coarse spatial grids in the boundary layer. Some energy stabilities of the numerical scheme are rigorously derived. Numerical examples illustrate the accuracy, stability and efficiency of the algorithm.
{"title":"Exponential time difference methods with spatial exponential approximations for solving boundary layer problems","authors":"Liyong Zhu, Xinwei Wang, Tianzheng Lu","doi":"10.1002/num.23145","DOIUrl":"https://doi.org/10.1002/num.23145","url":null,"abstract":"In this work, an efficient and stable exponential time difference method is presented for solving boundary layer problems. By combining exponential time difference schemes with spatial direct discontinuous Galerkin discretization based on exponential boundary layer approximations, the proposed algorithm not only may admit large time step sizes but also could provide good spatial approximations even if on rather coarse spatial grids in the boundary layer. Some energy stabilities of the numerical scheme are rigorously derived. Numerical examples illustrate the accuracy, stability and efficiency of the algorithm.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181162","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nonlocal models have demonstrated their indispensability in numerical simulations across a spectrum of critical domains, ranging from analyzing crack and fracture behavior in structural engineering to modeling anomalous diffusion phenomena in materials science and simulating convection processes in heterogeneous environments. In this study, we present a novel framework for constructing nonlocal convection–diffusion models using Gaussian‐type kernels. Our framework uniquely formulates the diffusion term by correlating the constant diffusion coefficient with the variance of the Gaussian kernel. Simultaneously, the convection term is defined by integrating the variable velocity field into the kernel as the expectation of a multivariate Gaussian distribution, facilitating a comprehensive representation of convective transport phenomena. We rigorously establish the well‐posedness of the proposed nonlocal model and derive a maximum principle to ensure its stability and reliability in numerical simulations. Furthermore, we develop a meshfree discretization scheme tailored for numerically simulating our model, designed to uphold both the discrete maximum principle and asymptotic compatibility. Through extensive numerical experiments, we validate the efficacy and versatility of our framework, demonstrating its superior performance compared to existing approaches.
{"title":"A nonlocal convection–diffusion model with Gaussian‐type kernels and meshfree discretization","authors":"Hao Tian, Xiaojuan Liu, Chenguang Liu, Lili Ju","doi":"10.1002/num.23141","DOIUrl":"https://doi.org/10.1002/num.23141","url":null,"abstract":"Nonlocal models have demonstrated their indispensability in numerical simulations across a spectrum of critical domains, ranging from analyzing crack and fracture behavior in structural engineering to modeling anomalous diffusion phenomena in materials science and simulating convection processes in heterogeneous environments. In this study, we present a novel framework for constructing nonlocal convection–diffusion models using Gaussian‐type kernels. Our framework uniquely formulates the diffusion term by correlating the constant diffusion coefficient with the variance of the Gaussian kernel. Simultaneously, the convection term is defined by integrating the variable velocity field into the kernel as the expectation of a multivariate Gaussian distribution, facilitating a comprehensive representation of convective transport phenomena. We rigorously establish the well‐posedness of the proposed nonlocal model and derive a maximum principle to ensure its stability and reliability in numerical simulations. Furthermore, we develop a meshfree discretization scheme tailored for numerically simulating our model, designed to uphold both the discrete maximum principle and asymptotic compatibility. Through extensive numerical experiments, we validate the efficacy and versatility of our framework, demonstrating its superior performance compared to existing approaches.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the asymptotical compatibility of the Fourier spectral method in multidimensional space for the nonlocal Ohta–Kawasaka model, a more generalized version of the model proposed in our previous work (Y. Zhao and W. Luo, Physica D 458 (2024), 133989). By introducing the Fourier collocation discretization for the spatial variable, we show that the asymptotical compatibility holds in 2D and 3D over a periodic domain. For the temporal discretization, we adopt the second‐order backward differentiation formula method. We prove that for certain nonlocal kernels, the proposed time discretization schemes inherit the energy dissipation law. In the numerical experiments, we verify the asymptotical compatibility, the second‐order temporal convergence rate, and the energy stability of the proposed schemes. More importantly, we discover a novel square lattice pattern when certain nonlocal kernel are applied in the model. In addition, our numerical experiments confirm the existence of an upper bound for the optimal number of bubbles in 2D for some specific nonlocal kernels. Finally, we numerically explore the promotion/demotion effect induced by the nonlocal horizon , which is consistent with the theoretical studies presented in our earlier work (Y. Zhao and W. Luo, Physica D 458 (2024), 133989).
我们研究了非局部 Ohta-Kawasaka 模型的傅立叶谱方法在多维空间的渐近相容性，该模型是我们之前工作（Y. Zhao and W. Luo, Physica D 458 (2024), 133989）中提出的模型的更广义版本。通过引入空间变量的傅立叶配位离散化，我们证明了在周期域上的二维和三维渐近相容性。对于时间离散化，我们采用了二阶反向微分公式法。我们证明，对于某些非局部核，所提出的时间离散化方案继承了能量耗散规律。在数值实验中，我们验证了所提方案的渐近相容性、二阶时间收敛率和能量稳定性。更重要的是，当在模型中应用某些非局部核时，我们发现了一种新颖的方格模式。此外，我们的数值实验证实，对于某些特定的非局部核，存在二维最佳气泡数量的上限。最后，我们用数值方法探讨了非局部水平线诱导的促进/移动效应，这与我们早期工作（Y. Zhao and W. Luo, Physica D 458 (2024), 133989）中的理论研究一致。
{"title":"Asymptotically compatible schemes for nonlocal Ohta–Kawasaki model","authors":"Wangbo Luo, Yanxiang Zhao","doi":"10.1002/num.23143","DOIUrl":"https://doi.org/10.1002/num.23143","url":null,"abstract":"We study the asymptotical compatibility of the Fourier spectral method in multidimensional space for the nonlocal Ohta–Kawasaka model, a more generalized version of the model proposed in our previous work (Y. Zhao and W. Luo, Physica D 458 (2024), 133989). By introducing the Fourier collocation discretization for the spatial variable, we show that the asymptotical compatibility holds in 2D and 3D over a periodic domain. For the temporal discretization, we adopt the second‐order backward differentiation formula method. We prove that for certain nonlocal kernels, the proposed time discretization schemes inherit the energy dissipation law. In the numerical experiments, we verify the asymptotical compatibility, the second‐order temporal convergence rate, and the energy stability of the proposed schemes. More importantly, we discover a novel square lattice pattern when certain nonlocal kernel are applied in the model. In addition, our numerical experiments confirm the existence of an upper bound for the optimal number of bubbles in 2D for some specific nonlocal kernels. Finally, we numerically explore the promotion/demotion effect induced by the nonlocal horizon , which is consistent with the theoretical studies presented in our earlier work (Y. Zhao and W. Luo, Physica D 458 (2024), 133989).","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a high accuracy numerical scheme for solving the two‐dimensional time‐fractional advection‐diffusion equation including mixed derivatives, where the variable‐step Alikhanov formula and a fourth‐order compact approximation are employed to time and space derivatives, respectively. Under mild assumptions on the time step‐sizes, we obtain the unconditional stability and high‐order convergence (second‐order in time and fourth‐order in space) of the proposed scheme by energy method. The theoretical statements are justified by the numerical experiments.
{"title":"A variable‐step high‐order scheme for time‐fractional advection‐diffusion equation with mixed derivatives","authors":"Junhong Feng, Pin Lyu, Seakweng Vong","doi":"10.1002/num.23140","DOIUrl":"https://doi.org/10.1002/num.23140","url":null,"abstract":"We consider a high accuracy numerical scheme for solving the two‐dimensional time‐fractional advection‐diffusion equation including mixed derivatives, where the variable‐step Alikhanov formula and a fourth‐order compact approximation are employed to time and space derivatives, respectively. Under mild assumptions on the time step‐sizes, we obtain the unconditional stability and high‐order convergence (second‐order in time and fourth‐order in space) of the proposed scheme by energy method. The theoretical statements are justified by the numerical experiments.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study a time splitting Fourier pseudo‐spectral (TSFP) method for the nonlinear Schrödinger equation with wave operator (NLSW). The nonlinear strength of the NLSW is characterized by . Specifically, we propose a coupled system which is equivalent to the NLSW and then apply the TSFP method to this system. As a geometric advantage, the TSFP method has time symmetry and conserves the discrete mass. Rigorous convergence analysis is provided to establish improved error bounds at up to the long‐time at where depends on the smoothness of the solution. Compared with the error bounds obtained by traditional analysis, our error bounds are greatly improved, especially when the problem presents weak nonlinearity, i.e. . In error analysis, combining with classical numerical analysis tools, we adopt the regularity compensation oscillation (RCO) technique to study the error accumulation process in detail and then establish the improved error bounds. The numerical experiments support our theoretical analysis. In addition, the numerical results show the long‐term stability of discrete energy.
{"title":"Improved error bounds on a time splitting method for the nonlinear Schrödinger equation with wave operator","authors":"Jiyong Li","doi":"10.1002/num.23139","DOIUrl":"https://doi.org/10.1002/num.23139","url":null,"abstract":"In this article, we study a time splitting Fourier pseudo‐spectral (TSFP) method for the nonlinear Schrödinger equation with wave operator (NLSW). The nonlinear strength of the NLSW is characterized by . Specifically, we propose a coupled system which is equivalent to the NLSW and then apply the TSFP method to this system. As a geometric advantage, the TSFP method has time symmetry and conserves the discrete mass. Rigorous convergence analysis is provided to establish improved error bounds at up to the long‐time at where depends on the smoothness of the solution. Compared with the error bounds obtained by traditional analysis, our error bounds are greatly improved, especially when the problem presents weak nonlinearity, i.e. . In error analysis, combining with classical numerical analysis tools, we adopt the regularity compensation oscillation (RCO) technique to study the error accumulation process in detail and then establish the improved error bounds. The numerical experiments support our theoretical analysis. In addition, the numerical results show the long‐term stability of discrete energy.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For boundary value problem of an elliptic equation with variable coefficients describing the physical field distribution in inhomogeneous media, the parametrix can represent the solution in terms of volume and surface potentials, with the drawback that the volume potential involving in the solution expression requires heavy computational costs as well as the solvability of the integral equations with respect to the density pair. We introduce an modified integral expression for the solution to an elliptic equation in divergence form under the parametrix framework. The well‐posedness of the linear integral system with respect to the density functions to be determined is rigorously proved. Based on the singularity decomposition for the parametrix, we propose two schemes to deal with the volume integrals so that the density functions can be solved efficiently. One method is an adaptive discretization scheme for computing the integrals with continuous integrands, leading to the uniform accuracy of the integrals in the whole domain, and consequently the efficient computations for the density functions. The other method is the dual reciprocity method which is a meshless approach converting the volume integrals into boundary integrals equivalently by expressing the volume density as the combination of the radial basis functions determined by the interior grids. The proposed schemes are justified numerically to be of satisfactory computation costs. Numerical examples in 2‐dimensional and 3‐dimensional cases are presented to show the validity of the proposed schemes.
{"title":"Numerical solution of the boundary value problem of elliptic equation by Levi function scheme","authors":"Jinchao Pan, Jijun Liu","doi":"10.1002/num.23142","DOIUrl":"https://doi.org/10.1002/num.23142","url":null,"abstract":"For boundary value problem of an elliptic equation with variable coefficients describing the physical field distribution in inhomogeneous media, the parametrix can represent the solution in terms of volume and surface potentials, with the drawback that the volume potential involving in the solution expression requires heavy computational costs as well as the solvability of the integral equations with respect to the density pair. We introduce an modified integral expression for the solution to an elliptic equation in divergence form under the parametrix framework. The well‐posedness of the linear integral system with respect to the density functions to be determined is rigorously proved. Based on the singularity decomposition for the parametrix, we propose two schemes to deal with the volume integrals so that the density functions can be solved efficiently. One method is an adaptive discretization scheme for computing the integrals with continuous integrands, leading to the uniform accuracy of the integrals in the whole domain, and consequently the efficient computations for the density functions. The other method is the dual reciprocity method which is a meshless approach converting the volume integrals into boundary integrals equivalently by expressing the volume density as the combination of the radial basis functions determined by the interior grids. The proposed schemes are justified numerically to be of satisfactory computation costs. Numerical examples in 2‐dimensional and 3‐dimensional cases are presented to show the validity of the proposed schemes.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":3.9,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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