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1.
A fully discrete discontinuous Galerkin method is introduced for solving
time-dependent Maxwell's equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in
our scheme, discontinuous Galerkin methods are used to discretize not only the spatial
domain but also the temporal domain. The proposed numerical scheme is proved to be
unconditionally stable, and a convergent rate $\mathcal{O}((∆t)^{r+1}+h^{k+1/2})$ is established under the $L^2$ -norm when polynomials of degree at most $r$ and $k$ are used for temporal and
spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order $(∆t)^{2r+1}$ in
time step is observed numerically for the numerical fluxes w.r.t. temporal variable at
the grid points. 相似文献
2.
An Efficient Finite Element Method with Exponential Mesh Refinement for the Solution of the Allen-Cahn Equation in Non-Convex Polygons
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Emine Celiker & Ping Lin 《Communications In Computational Physics》2020,28(4):1536-1560
In this paper we consider the numerical solution of the Allen-Cahn type
diffuse interface model in a polygonal domain. The intersection of the interface with
the re-entrant corners of the polygon causes strong corner singularities in the solution.
To overcome the effect of these singularities on the accuracy of the approximate solution, for the spatial discretization we develop an efficient finite element method with
exponential mesh refinement in the vicinity of the singular corners, that is based on
($k$−1)-th order Lagrange elements, $k$≥2 an integer. The problem is fully discretized by
employing a first-order, semi-implicit time stepping scheme with the Invariant Energy
Quadratization approach in time, which is an unconditionally energy stable method.
It is shown that for the error between the exact and the approximate solution, an accuracy of $\mathcal{O}$($h^k$+$τ$) is attained in the $L^2$-norm for the number of $\mathcal{O}$($h^{−2}$ln$h^{−1}$) spatial
elements, where $h$ and $τ$ are the mesh and time steps, respectively. The numerical
results obtained support the analysis made. 相似文献
3.
Zuowei Shen Haizhao Yang & Shijun Zhang 《Communications In Computational Physics》2020,28(5):1768-1811
This paper quantitatively characterizes the approximation power of deep
feed-forward neural networks (FNNs) in terms of the number of neurons. It is shown
by construction that ReLU FNNs with width$\mathcal{O}$(max{$d⌊N^{1/d}⌋$,$N$+1}) and depth $\mathcal{O}(L)$ can approximate an arbitrary Hölder continuous function of order $α∈(0,1]$ on $[0,1]^d$ with a nearly tight approximation rate $\mathcal{O}(\sqrt{d}N^{−2α/d}L^{−2α/d})$ measured in $L^p$ -norm for
any $N,L∈\mathbb{N}^+$ and $p∈[1,∞]$. More generally for an arbitrary continuous function $f$ on $[0,1]^d$ with a modulus of continuity $ω_f
(·)$, the constructive approximation rate
is $\mathcal{O}(\sqrt{d}ω_f(N^{−2α/d}L^{−2α/d}))$. We also extend our analysis to $f$ on irregular domains or
those localized in an ε-neighborhood of a $d_\mathcal{M}$-dimensional smooth manifold $\mathcal{M}⊆[0,1]^d$ with $d_\mathcal{M}≪d$. Especially, in the case of an essentially low-dimensional domain, we
show an approximation rate $\mathcal{O}(ω_f(\frac{ε}{1−δ}\sqrt{\frac{d}{d_δ}}+ε)+\sqrt{d}ω_f(\frac{\sqrt{d}}{1−δ\sqrt{d_δ}}N^{−2α/d_δ}L^{−2α/d_δ})$ for
ReLU FNNs to approximate $f$ in the $ε$-neighborhood, where $d_δ=\mathcal{O}(d_\mathcal{M}\frac{\rm{ln}(d/δ)}{δ^2})$ for any $δ∈(0,1)$ as a relative error for a projection to approximate an isometry when projecting $\mathcal{M}$ to a $d_δ$-dimensional domain. 相似文献
4.
A Mixed Finite Element Scheme for Biharmonic Equation with Variable Coefficient and von Kármán Equations
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Huangxin Chen Amiya K. Pani & Weifeng Qiu 《Communications In Computational Physics》2022,31(5):1434-1466
In this paper, a new mixed finite element scheme using element-wise stabilization is introduced for the biharmonic equation with variable coefficient on Lipschitz polyhedral domains. The proposed scheme doesn’t involve any integration along
mesh interfaces. The gradient of the solution is approximated by $H$(div)-conforming $BDM_{k+1}$ element or vector valued Lagrange element with order $k+1,$ while the solution is approximated by Lagrange element with order $k+2$ for any $k≥0.$ This scheme
can be easily implemented and produces symmetric and positive definite linear system. We provide a new discrete $H^2$-norm stability, which is useful not only in analysis
of this scheme but also in ${\rm C}^0$ interior penalty methods and DG methods. Optimal convergences in both discrete $H^2$-norm and $L^2$-norm are derived. This scheme with its
analysis is further generalized to the von Kármán equations. Finally, numerical results
verifying the theoretical estimates of the proposed algorithms are also presented. 相似文献
5.
Fast One-Dimensional Convolution with General Kernels Using Sum-of-Exponential Approximation
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Yong Zhang Chijie Zhuang & Shidong Jiang 《Communications In Computational Physics》2021,29(5):1570-1582
Based on the recently-developed sum-of-exponential (SOE) approximation,
in this article, we propose a fast algorithm to evaluate the one-dimensional convolution potential $φ(x)=K∗ρ=∫^1_{0}K(x−y)ρ(y)dy$ at (non)uniformly distributed target grid
points {$x_i$}$^M_{i=1}$, where the kernel $K(x)$ might be singular at the origin and the source
density function $ρ(x)$ is given on a source grid ${{{y_i}}}^N_{j=1}$ which can be different from
the target grid. It achieves an optimal accuracy, inherited from the interpolation of
the density $ρ(x)$, within $\mathcal{O}(M+N)$ operations. Using the kernel's SOE approximation $K_{ES}$, the potential is split into two integrals: the exponential convolution $φ_{ES}$=$K_{ES}∗ρ$ and the local correction integral $φ_{cor}=(K−K_{ES})∗ρ$. The exponential convolution is
evaluated via the recurrence formula that is typical of the exponential function. The
local correction integral is restricted to a small neighborhood of the target point where
the kernel singularity is considered. Rigorous estimates of the optimal accuracy are
provided. The algorithm is ideal for parallelization and favors easy extensions to complicated kernels. Extensive numerical results for different kernels are presented. 相似文献
6.
Xiaoqiang Yue Shi Shu Xiaowen Xu & Zhiyang Zhou 《Communications In Computational Physics》2015,18(5):1313-1335
The paper aims to develop an effective preconditioner and conduct the convergence
analysis of the corresponding preconditioned GMRES for the solution of discrete
problems originating from multi-group radiation diffusion equations. We firstly
investigate the performances of the most widely used preconditioners (ILU(k) and
AMG) and their combinations ($B_{co}$ and$\widetilde{B}_{co}$), and provide drawbacks on their feasibilities.
Secondly, we reveal the underlying complementarity of ILU(k) and AMG by analyzing
the features suitable for AMG using more detailed measurements on multiscale
nature of matrices and the effect of ILU(k) on multiscale nature. Moreover, we present
an adaptive combined preconditioner $B^α_{co}$ involving an improved ILU(0) along with
its convergence constraints. Numerical results demonstrate that $B^α_{co}$-GMRES holds the
best robustness and efficiency. At last, we analyze the convergence of GMRES with
combined preconditioning which not only provides a persuasive support for our proposed
algorithms, but also updates the existing estimation theory on condition numbers
of combined preconditioned systems. 相似文献
7.
Correlation Functions,Universal Ratios and Goldstone Mode Singularities in $n$-Vector Models
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J. Kaupužs R. V. N. Melnik & J. Rim&scaron āns 《Communications In Computational Physics》2014,15(5):1407-1430
Correlation functions in the $\mathcal{O}$$(n)$ models below the critical temperature are
considered. Based on Monte Carlo (MC) data, we confirm the fact stated earlier by Engels and Vogt, that the transverse two-plane correlation function of the $\mathcal{O}$$(4)$ model for
lattice sizes about $L=120$ and small external fields $h$ is very well described by a Gaussian approximation. However, we show that fits of not lower quality are provided by
certain non-Gaussian approximation. We have also tested larger lattice sizes, up to $L=512$. The Fourier-transformed transverse and longitudinal two-point correlation
functions have Goldstone mode singularities in the thermodynamic limit at $k→0$ and $h=+0$, i.e., $G_⊥$(k)$≃ak^{−λ_⊥}$ and$G_{||}($k$)≃bk^{−λ_{||}}$, respectively. Here $a$ and $b$ are the amplitudes, $k$=|k| is the magnitude of the wave vector k. The exponents $λ_⊥$, $λ_{||}$ and the
ratio $bM^2/a^2$, where $M$ is the spontaneous magnetization, are universal according to
the GFD (grouping of Feynman diagrams) approach. Here we find that the universality follows also from the standard (Gaussian) theory, yielding $bM^2/a^2$=$(n−1)/16$. Our
MC estimates of this ratio are $0.06±0.01$ for $n=2$, $0.17±0.01$ for $n=4$ and $0.498±0.010$ for $n=10$. According to these and our earlier MC results, the asymptotic behavior and
Goldstone mode singularities are not exactly described by the standard theory. This is
expected from the GFD theory. We have found appropriate analytic approximations
for $G_⊥$(k) and $G_{||}$(k), well fitting the simulation data for small $k$. We have used them
to test the Patashinski-Pokrovski relation and have found that it holds approximately. 相似文献
8.
An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems
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Ye Li 《Communications In Computational Physics》2016,19(2):442-472
In this paper, we propose a uniformly convergent adaptive finite element
method with hybrid basis (AFEM-HB) for the discretization of singularly perturbed
nonlinear eigenvalue problems under constraints with applications in Bose-Einstein
condensation (BEC) and quantum chemistry. We begin with the time-independent
Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed
nonlinear eigenvalue problem under a constraint. Matched asymptotic approximations
for the problem are reviewed to confirm the asymptotic behaviors of the solutions
in the boundary/interior layer regions. By using the normalized gradient flow, we
propose an adaptive finite element with hybrid basis to solve the singularly perturbed
nonlinear eigenvalue problem. Our basis functions and the mesh are chosen adaptively
to the small parameter ε. Extensive numerical results are reported to show the
uniform convergence property of our method. We also apply the AFEM-HB to compute
the ground and excited states of BEC with box/harmonic/optical lattice potential
in the semiclassical regime (0<ε≪1). In addition, we give a detailed error analysis of
our AFEM-HB to a simpler singularly perturbed two point boundary value problem,
show that our method has a minimum uniform convergence order $\mathcal{O}$(1/$(NlnN)^\frac{2}{3}$). 相似文献
9.
G. J. I. M. van der Werf A. J. Tonino 《Archives of orthopaedic and trauma surgery》1984,103(2):128-130
Summary Eleven posterior tibial tendon transfers were performed in eight patients with spastic equinovarus with an average follow-up of 3 years (range
years). The main goals—to eliminate the need for braces and to improve gait—were achieved in all patients. An adequate operation technique kept the well-known postoperative complications, e.g., valgus and calcaneal deformity, to a minimum.
Zusammenfassung Die Transposition der Musc. tibialis posterior-Sehne als operative Behandlung des spastischen Equinovarus wurde elfmal durchgeführt bei acht Patienten. Die Ergebnisse wurden Jahre nach der Operation kontrolliert. Das Hauptziel: Unabhängigkeit von Orthesen and Verbesserung der Gehfähigkeit, konnte bei allen Patienten erreicht werden. Eine adequate Operationstechnik konnte die bekannten postoperativen Komplikationen, wie eine Valgus- und Hackenfuß-Deformität bis auf ein Minimum beschränken.相似文献
10.
The validity of using mechanical measures of work to indicate the metabolic energy consumption during normal gait was examined. These mechanical measures were (a) mechanical work done on the center of mass per kilogram body mass per second (\documentclass{article}\pagestyle{empty}\begin{document}$ \dot W_{{\rm cm}} $\end{document}), calculated by integration of ground reaction forces measured by force platforms; (b) total body segmental work per kilogram body mass per second (\documentclass{article}\pagestyle{empty}\begin{document}$ \dot W_{{\rm seg}} $\end{document}), calculated from individual body segment energies measured by motion analysis; and (c) the sum of the normalized absolute moment impulses per second acting on the joints of the lower extremities (\documentclass{article}\pagestyle{empty}\begin{document}$ \dot M $\end{document}), calculated from both force and motion data. The metabolic energy consumption, determined by analysis of expired air, and the three mechanical measures of work were calculated for six normal subjects walking at five speeds. Each measure of mechanical work per second walked was highly correlated with metabolic energy consumption/kg · s (r = 0.89 for Wcm, r = 0.79 for \documentclass{article}\pagestyle{empty}\begin{document}$ \dot W_{{\rm seg}} $\end{document}, and r = 0.85 for M), but a poorer correlation was found between each measure of mechanical work per meter walked and net metabolic energy consumption/kg. m (r = 0.54 for Wcm, r = 0.28 for Wseg, and r = 0.03 for M). These mechanical parameters, particularly when measured per time, may be useful in comparing metabolic energy consumption between individuals or between different walking conditions for the same individual. 相似文献
11.
A New Family of Nonconforming Elements with $H$(curl)-Continuity for the 3D Quad-Curl Problem
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Baiju Zhang & Zhimin Zhang 《Communications In Computational Physics》2023,33(4):1069-1089
We propose and analyze a new family of nonconforming finite elements
for the three-dimensional quad-curl problem. The proposed finite element spaces are
subspaces of $\boldsymbol{H}$(curl), but not of $\boldsymbol{H}$(grad curl), which are different from the existing
nonconforming ones [10,12,13]. The well-posedness of the discrete problem is proved
and optimal error estimates in discrete $\boldsymbol{H}$(grad curl) norm, $\boldsymbol{H}$(curl) norm and $L^2$ norm
are derived. Numerical experiments are provided to illustrate the good performance
of the method and confirm our theoretical predictions. 相似文献
12.
Jinchao Xu 《Communications In Computational Physics》2020,28(5):1707-1745
We study a family of $H^m$-conforming piecewise polynomials based on the
artificial neural network, referred to as the finite neuron method (FNM), for numerical
solution of $2m$-th-order partial differential equations in$\mathbb{R}^d$ for any $m,d≥1$ and then
provide convergence analysis for this method. Given a general domain Ω$⊂\mathbb{R}^d$ and a
partition$\mathcal{T}_h$ of Ω, it is still an open problem in general how to construct a conforming finite element subspace of $H^m$(Ω) that has adequate approximation properties. By using
techniques from artificial neural networks, we construct a family of $H^m$-conforming
functions consisting of piecewise polynomials of degree $k$ for any $k≥m$ and we further obtain the error estimate when they are applied to solve the elliptic boundary
value problem of any order in any dimension. For example, the error estimates that $‖u−u_N‖_{H^m(\rm{Ω})}=\mathcal{O}(N^{−\frac{1}{2}−\frac{1}{d}})$ is obtained for the error between the exact solution $u$ and
the finite neuron approximation $u_N$. A discussion is also provided on the difference
and relationship between the finite neuron method and finite element methods (FEM).
For example, for the finite neuron method, the underlying finite element grids are not
given a priori and the discrete solution can be obtained by only solving a non-linear
and non-convex optimization problem. Despite the many desirable theoretical properties of the finite neuron method analyzed in the paper, its practical value requires
further investigation as the aforementioned underlying non-linear and non-convex optimization problem can be expensive and challenging to solve. For completeness and
the convenience of the reader, some basic known results and their proofs are introduced. 相似文献
13.
Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System
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Yong Zhang 《Communications In Computational Physics》2013,13(5):1357-1388
We study compact finite difference methods for the Schrödinger-Poisson
equation in a bounded domain and establish their optimal error estimates under proper
regularity assumptions on wave function $ψ$ and external potential $V(x)$. The Crank-Nicolson compact finite difference method and the semi-implicit compact finite difference method are both of order $\mathcal{O}$($h^4$+$τ^2$) in discrete $l^2$, $H^1$ and $l^∞$ norms with mesh
size $h$ and time step $τ$. For the errors of compact finite difference approximation to
the second derivative and Poisson potential are nonlocal, thus besides the standard
energy method and mathematical induction method, the key technique in analysis is
to estimate the nonlocal approximation errors in discrete $l^∞$ and $H^1$ norm by discrete
maximum principle of elliptic equation and properties of some related matrix. Also
some useful inequalities are established in this paper. Finally, extensive numerical results are reported to support our error estimates of the numerical methods. 相似文献
14.
A Comparison of Higher-Order Weak Numerical Schemes for Stopped Stochastic Differential Equations
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Francisco Bernal & Juan A. Acebró n 《Communications In Computational Physics》2016,20(3):703-732
We review, implement, and compare numerical integration schemes for spatially
bounded diffusions stopped at the boundary which possess a convergence rate of
the discretization error with respect to the time step $h$ higher than $\mathcal{O}$$(√h)$. We address
specific implementation issues of the most general-purpose of such schemes. They
have been coded into a single Matlab program and compared, according to their accuracy
and computational cost, on a wide range of problems in up to R48. The paper is
self-contained and the code will be made freely downloadable. 相似文献
15.
Dominik Dierkes Florian Kummer & Dominik Plü macher 《Communications In Computational Physics》2021,30(1):288-320
We present a high-order discontinuous Galerkin (DG) scheme to solve the
system of helically symmetric Navier-Stokes equations which are discussed in [28].
In particular, we discretize the helically reduced Navier-Stokes equations emerging
from a reduction of the independent variables such that the remaining variables are: $t$, $r$, $ξ$ with $ξ=az+bϕ$, where $r$, $ϕ$, $z$ are common cylindrical coordinates and $t$ the
time. Beside this, all three velocity components are kept non-zero. A new non-singular
coordinate $η$ is introduced which ensures that a mapping of helical solutions into the
three-dimensional space is well defined. Using that, periodicity conditions for the
helical frame as well as uniqueness conditions at the centerline axis $r=0$ are derived. In
the sector near the axis of the computational domain a change of the polynomial basis
is implemented such that all physical quantities are uniquely defined at the centerline.For the temporal integration, we present a semi-explicit scheme of third order
where the full spatial operator is split into a Stokes operator which is discretized
implicitly and an operator for the nonlinear terms which is treated explicitly. Computations are conducted for a cylindrical shell, excluding the centerline axis, and for
the full cylindrical domain, where the centerline is included. In all cases we obtain the
convergence rates of order $\mathcal{O}(h^{k+1})$ that are expected from DG theory.In addition to the first DG discretization of the system of helically invariant Navier-Stokes equations, the treatment of the central axis, the resulting reduction of the DG
space, and the simultaneous use of a semi-explicit time stepper are of particular novelty. 相似文献
16.
On the Effect of Ghost Force in the Quasicontinuum Method: Dynamic Problems in One Dimension
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Numerical error caused by "ghost forces" in a quasicontinuum method is
studied in the context of dynamic problems. The error in the discrete W1,∞ norm is analyzed for the time scale $\mathcal{O}$($ε$) and the time scale $\mathcal{O}$(1) with ε being the lattice spacing. 相似文献
17.
Directional $\mathcal{H}^2$ Compression Algorithm: Optimisations and Application to a Discontinuous Galerkin BEM for the Helmholtz Equation
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Nadir-Alexandre Messaï Sebastien Pernet & Abdesselam Bouguerra 《Communications In Computational Physics》2022,31(5):1585-1635
This study aimed to specialise a directional $\mathcal{H}^2
(\mathcal{D}\mathcal{H}^2)$ compression to matrices arising from the discontinuous Galerkin (DG) discretisation of the hypersingular
equation in acoustics. The significant finding is an algorithm that takes a DG stiffness matrix and finds a near-optimal $\mathcal{D}\mathcal{H}^2$ approximation for low and high-frequency
problems. We introduced the necessary special optimisations to make this algorithm
more efficient in the case of a DG stiffness matrix. Moreover, an automatic parameter
tuning strategy makes it easy to use and versatile. Numerical comparisons with a classical Boundary Element Method (BEM) show that a DG scheme combined with a $\mathcal{D}\mathcal{H}^2$ gives better computational efficiency than a classical BEM in the case of high-order finite elements and $hp$ heterogeneous meshes. The results indicate that DG is suitable
for an auto-adaptive context in integral equations. 相似文献
18.
Numerical Solution of 3D Poisson-Nernst-Planck Equations Coupled with Classical Density Functional Theory for Modeling Ion and Electron Transport in a Confined Environment
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Da Meng Bin Zheng Guang Lin & Maria L. Sushko 《Communications In Computational Physics》2014,16(5):1298-1322
We have developed efficient numerical algorithms for solving 3D steady-state
Poisson-Nernst-Planck (PNP) equations with excess chemical potentials described
by the classical density functional theory (cDFT). The coupled PNP equations are discretized
by a finite difference scheme and solved iteratively using the Gummel method
with relaxation. The Nernst-Planck equations are transformed into Laplace equations
through the Slotboom transformation. Then, the algebraic multigrid method is
applied to efficiently solve the Poisson equation and the transformed Nernst-Planck
equations. A novel strategy for calculating excess chemical potentials through fast
Fourier transforms is proposed, which reduces computational complexity from $\mathcal{O}$($N^2$) to $\mathcal{O}$($NlogN$), where $N$ is the number of grid points. Integrals involving the Dirac
delta function are evaluated directly by coordinate transformation, which yields more
accurate results compared to applying numerical quadrature to an approximated delta
function. Numerical results for ion and electron transport in solid electrolyte for lithium-ion
(Li-ion) batteries are shown to be in good agreement with the experimental data
and the results from previous studies. 相似文献
19.
Lei Wang Robert Krasny & Svetlana Tlupova 《Communications In Computational Physics》2020,28(4):1415-1436
A kernel-independent treecode (KITC) is presented for fast summation of
particle interactions. The method employs barycentric Lagrange interpolation at
Chebyshev points to approximate well-separated particle-cluster interactions. The
KITC requires only kernel evaluations, is suitable for non-oscillatory kernels, and relies on the scale-invariance property of barycentric Lagrange interpolation. For a given
level of accuracy, the treecode reduces the operation count for pairwise interactions
from $\mathcal{O}$($N^2$) to $\mathcal{O}$($N$log$N$), where $N$ is the number of particles in the system. The algorithm is demonstrated for systems of regularized Stokeslets and rotlets in 3D, and
numerical results show the treecode performance in terms of error, CPU time, and
memory consumption. The KITC is a relatively simple algorithm with low memory
consumption, and this enables a straightforward OpenMP parallelization. 相似文献
20.
We describe our implementation of a parallel fast multipole method for evaluating
potentials for discrete and continuous source distributions. The first requires
summation over the source points and the second requiring integration over a continuous
source density. Both problems require$\mathcal{O}$($N^2$) complexity when computed directly;
however, can be accelerated to $\mathcal{O}$($N$) time using FMM. In our PVFMM software
library, we use kernel independent FMM and this allows us to compute potentials for
a wide range of elliptic kernels. Our method is high order, adaptive and scalable. In
this paper, we discuss several algorithmic improvements and performance optimizations
including cache locality, vectorization, shared memory parallelism and use of
coprocessors. Our distributed memory implementation uses space-filling curve for
partitioning data and a hypercube communication scheme. We present convergence
results for Laplace, Stokes and Helmholtz (low wavenumber) kernels for both particle
and volume FMM. We measure efficiency of our method in terms of CPU cycles per
unknown for different accuracies and different kernels. We also demonstrate scalability
of our implementation up to several thousand processor cores on the Stampede
platform at the Texas Advanced Computing Center. 相似文献