Among various algorithms of multifractal analysis (MFA) for complex networks, the sandbox MFA algorithm behaves with the best computational efficiency. However, the existing sandbox algorithm is still computationally ...
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Among various algorithms of multifractal analysis (MFA) for complex networks, the sandbox MFA algorithm behaves with the best computational efficiency. However, the existing sandbox algorithm is still computationally expensive for MFA of large-scale networks with tens of millions of nodes. It is also not clear whether MFA results can be improved by a largely increased size of a theoretical network. To tackle these challenges, a computationally efficient sandbox algorithm (CESA) is presented in this paper for MFA of large-scale networks. Distinct from the existing sandbox algorithm that uses the shortest-path distance matrix to obtain the required information for MFA of networks, our CESA employs the compressed sparse row format of the adjacency matrix and the breadth-first search technique to directly search the neighbor nodes of each layer of center nodes, and then to retrieve the required information. A theoretical analysis reveals that the CESA reduces the time complexity of the existing sandbox algorithm from cubic to quadratic, and also improves the space complexity from quadratic to linear. Then the CESA is demonstrated to be effective, efficient, and feasible through the MFA results of (u,v)-flower model networks from the fifth to the 12th generations. It enables us to study the multifractality of networks of the size of about 11 million nodes with a normal desktop computer. Furthermore, we have also found that increasing the size of (u,v)-flower model network does improve the accuracy of MFA results. Finally, our CESA is applied to a few typical real-world networks of large scale.
The Hm-conforming virtual elements of any degree k on any shape of polytope in n with m, n ≥ 1 and k ≥ m are recursively constructed by gluing conforming virtual elements on faces in a universal way. For the lowest ...
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This paper focuses on designing a lowest-order divergence-free virtual element method for solving Navier-Stokes equations with a nonlinear damping term on polygonal meshes. The exact divergence-free property of virtua...
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This paper focuses on designing a lowest-order divergence-free virtual element method for solving Navier-Stokes equations with a nonlinear damping term on polygonal meshes. The exact divergence-free property of virtual space preserves the mass-conservation of the system. With the application of Helmholtz projection, we provide stability estimates regarding the velocity. An optimal convergence estimate is derived, showing that the error estimate for the velocity in energy norm is pressure-independent. Finally, we perform various numerical simulations to validate the accuracy of our theoretical findings.
Due to the successful applications in engineering,physics,biology,finance,etc.,there has been substantial interest in fractional diffusion equations over the past few decades,and literatures on developing and analyzin...
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Due to the successful applications in engineering,physics,biology,finance,etc.,there has been substantial interest in fractional diffusion equations over the past few decades,and literatures on developing and analyzing efficient and accurate numerical methods for reliably simulating such equations are vast and fast *** paper gives a concise overview on finite element methods for these equations,which are divided into time fractional,space fractional and time-space fractional diffusion ***,we also involve some relevant topics on the regularity theory,the well-posedness,and the fast algorithm.
This paper considers a class of discontinuous Galerkin method,which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis,for numerically solving nonautonomous Stratonovich stochastic delay dif...
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This paper considers a class of discontinuous Galerkin method,which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis,for numerically solving nonautonomous Stratonovich stochastic delay differential *** prove that the discontinuous Galerkin scheme is strongly convergent:globally stable and analogously asymptotically stable in mean square *** addition,this method can be easily extended to solve nonautonomous Stratonovich stochastic pantograph differential *** tests indicate that the method has first-order and half-order strong mean square convergence,when the diffusion term is without delay and with delay,respectively.
This article focuses on the development of high-order energy stable schemes for the multi-length-scale incommensurate phase-field crystal model which is able to study the phase behavior of aperiodic structures. These ...
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In this paper, we develop a high-order adaptive virtual element method (VEM) to simulate the self-consistent field theory (SCFT) model in arbitrary domains. The VEM is very flexible in handling general polygon element...
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In this paper, we develop a residual-type a posteriori error estimation for an interior penalty virtual element method (IPVEM) for the Kirchhoff plate bending problem. Building on the work in Feng and Yu (2024), we ad...
In this paper, we develop a residual-type a posteriori error estimation for an interior penalty virtual element method (IPVEM) for the Kirchhoff plate bending problem. Building on the work in Feng and Yu (2024), we adopt a modified discrete variational formulation that incorporates the H 1 -elliptic projector in the jump and average terms. This allows us to simplify the numerical implementation by including the H 1 -elliptic projector in the computable error estimators. We derive the reliability and efficiency of the a posteriori error bound by constructing an enriching operator and establishing some related error estimates that align with C 0 -continuous interior penalty finite element methods. As observed in the a priori analysis, the interior penalty virtual elements exhibit similar behaviors to C 0 -continuous elements despite its H 1 -nonconforming. This observation extends to the a posteriori estimate since we do not need to account for the jumps of the function itself in the discrete scheme and the error estimators. As an outcome of the error estimator, an adaptive VEM is introduced by means of the mesh refinement strategy with the one-hanging-node rule. Numerical results from several benchmark tests confirm the robustness of the proposed error estimators and show the efficiency of the resulting adaptive VEM.
In this paper, by modifying the smoothness factor of the third-order CWENO scheme, we present the two-parameter CWENO-NZ3 scheme to improve accuracy at critical points. We selected some classical examples for numerica...
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This paper aims to develop an efficient adaptive finite element method for the second-order elliptic problem. Although the theory for adaptive finite element methods based on residual-type a posteriori error estimator...
This paper aims to develop an efficient adaptive finite element method for the second-order elliptic problem. Although the theory for adaptive finite element methods based on residual-type a posteriori error estimator and bisection refinement has been well established, in practical computations, the use of non-asymptotic exactness of error estimator and the excessive number of adaptive iteration steps often lead to inefficiency of the adaptive algorithm. We propose an efficient adaptive finite element method based on high-accuracy techniques including the superconvergence recovery technique and high-quality mesh optimization. The centroidal Voronoi Delaunay triangulation mesh optimization is embedded in the mesh adaptation to provide a high-quality mesh, and then ensure the superconvergence property of the recovered gradient and the asymptotic exactness of the error estimator. A tailored adaptive strategy, which can generate high-quality meshes with a target number of vertices, is developed to ensure the adaptive computation process terminates within 7 steps. The effectiveness and robustness of the adaptive algorithm are numerically demonstrated.
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