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1.
Consider the inverse diffraction problem to determine a two-dimensional periodic structure from scattered elastic waves measured above the structure. We formulate the inverse problem as a least squares optimization problem, following the two-step algorithm by G. Bruckner and J. Elschner [Inverse Probl., 19 (2003), 315–329] for electromagnetic diffraction gratings. Such a method is based on the Kirsch-Kress optimization scheme and consists of two parts: a linear severely ill-posed problem and a nonlinear well-posed one. We apply this method to both smooth (C2) and piecewise linear gratings for the Dirichlet boundary value problem of the Navier equation. Numerical reconstructions from exact and noisy data illustrate the feasibility of the method. 相似文献
2.
A Bilinear Immersed Finite Volume Element Method for the Diffusion Equation with Discontinuous Coefficient
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This paper is to present a finite volume element (FVE) method based on the
bilinear immersed finite element (IFE) for solving the boundary value problems of the
diffusion equation with a discontinuous coefficient (interface problem). This method
possesses the usual FVE method's local conservation property and can use a structured
mesh or even the Cartesian mesh to solve a boundary value problem whose coefficient
has discontinuity along piecewise smooth nontrivial curves. Numerical examples are
provided to demonstrate features of this method. In particular, this method can produce
a numerical solution to an interface problem with the usual O(h2) (in L2 norm)
and O(h) (in H1 norm) convergence rates. 相似文献
3.
An Iterative Discontinuous Galerkin Method for Solving the Nonlinear Poisson Boltzmann Equation
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Peimeng Yin Yunqing Huang & Hailiang Liu 《Communications In Computational Physics》2014,16(2):491-515
An iterative discontinuous Galerkin (DG) method is proposed to solve the
nonlinear Poisson Boltzmann (PB) equation. We first identify a function space in which
the solution of the nonlinear PB equation is iteratively approximated through a series
of linear PB equations, while an appropriate initial guess and a suitable iterative parameter
are selected so that the solutions of linear PB equations are monotone within
the identified solution space. For the spatial discretization we apply the direct discontinuous
Galerkin method to those linear PB equations. More precisely, we use one
initial guess when the Debye parameter λ=O(1), and a special initial guess for λ≪1
to ensure convergence. The iterative parameter is carefully chosen to guarantee the existence,
uniqueness, and convergence of the iteration. In particular, iteration steps can
be reduced for a variable iterative parameter. Both one and two-dimensional numerical
results are carried out to demonstrate both accuracy and capacity of the iterative
DG method for both cases of λ=O(1) and λ≪1. The (m+1)th order of accuracy for
L2 and mth order of accuracy for H1for Pm elements are numerically obtained. 相似文献
4.
Mengjiao Jiao Yingda Cheng Yong Liu & Mengping Zhang 《Communications In Computational Physics》2020,28(3):927-966
In this paper, we develop central discontinuous Galerkin (CDG) finite element methods for solving the generalized Korteweg-de Vries (KdV) equations in one
dimension. Unlike traditional discontinuous Galerkin (DG) method, the CDG methods evolve two approximate solutions defined on overlapping cells and thus do not
need numerical fluxes on the cell interfaces. Several CDG schemes are constructed, including the dissipative and non-dissipative versions. L2error estimates are established
for the linear and nonlinear equation using several projections for different parameter
choices. Although we can not provide optimal a priori error estimate, numerical examples show that our scheme attains the optimal (k+1)-th order of accuracy when using
piecewise k-th degree polynomials for many cases. 相似文献
5.
Weak Galerkin and Continuous Galerkin Coupled Finite Element Methods for the Stokes-Darcy Interface Problem
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Hui Peng Qilong Zhai Ran Zhang & Shangyou Zhang 《Communications In Computational Physics》2020,28(3):1147-1175
We consider a model of coupled free and porous media flow governed by
Stokes equation and Darcy's law with the Beavers-Joseph-Saffman interface condition.
In this paper, we propose a new numerical approach for the Stokes-Darcy system. The
approach employs the classical finite element method for the Darcy region and the
weak Galerkin finite element method for the Stokes region. We construct corresponding discrete scheme and prove its well-posedness. The estimates for the corresponding numerical approximation are derived. Finally, we present some numerical experiments to validate the efficiency of the approach for solving this problem. 相似文献
6.
Pseudostress-Based Mixed Finite Element Methods for the Stokes Problem in Rn with Dirichlet Boundary Conditions I: A Priori Error Analysis
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Gabriel N. Gatica Antonio Má rquez & Manuel A. Sá nchez 《Communications In Computational Physics》2012,12(1):109-134
We consider a non-standard mixed method for the Stokes problem in Rn, n∈{2,3}, with Dirichlet boundary conditions, in which, after using the incompressibility condition to eliminate the pressure, the pseudostress tensor σ and the velocity vectorubecome the only unknowns. Then, we apply the Babuška-Brezzi theory to prove the well-posedness of the corresponding continuous and discrete formulations. In particular, we show that Raviart-Thomas elements of order k≥0 for σ and piecewise polynomials of degree k foruensure unique solvability and stability of the associated Galerkin scheme. In addition, we introduce and analyze an augmented approach for our pseudostress-velocity formulation. The methodology employed is based on the introduction of the Galerkin least-squares type terms arising from the constitutive and equilibrium equations, and the Dirichlet boundary condition for the velocity, all of them multiplied by suitable stabilization parameters. We show that these parameters can be chosen so that the resulting augmented variational formulation is defined by a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces. For instance, Raviart-Thomas elements of order k≥0 for σ and continuous piecewise polynomials of degree k+1 forubecome a feasible choice in this case. Finally, become a feasible choice in this case. Finally, extensive numerical experiments illustrating the good performance of the methods and comparing them with other procedures available in the literature, are provided. 相似文献
7.
A Sparse Grid Discrete Ordinate Discontinuous Galerkin Method for the Radiative Transfer Equation
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Jianguo Huang & Yue Yu 《Communications In Computational Physics》2021,30(4):1009-1036
The radiative transfer equation is a fundamental equation in transport theory and applications, which is a 5-dimensional PDE in the stationary one-velocity case,
leading to great difficulties in numerical simulation. To tackle this bottleneck, we first
use the discrete ordinate technique to discretize the scattering term, an integral with respect to the angular variables, resulting in a semi-discrete hyperbolic system. Then, we
make the spatial discretization by means of the discontinuous Galerkin (DG) method
combined with the sparse grid method. The final linear system is solved by the block
Gauss-Seidal iteration method. The computational complexity and error analysis are
developed in detail, which show the new method is more efficient than the original
discrete ordinate DG method. A series of numerical results are performed to validate
the convergence behavior and effectiveness of the proposed method. 相似文献
8.
Mingze Qin Ruishu Wang Qilong Zhai & Ran Zhang 《Communications In Computational Physics》2023,33(2):568-595
The weak Galerkin (WG) method is a nonconforming numerical method for
solving partial differential equations. In this paper, we introduce the WG method for
elliptic equations with Newton boundary condition in bounded domains. The Newton
boundary condition is a nonlinear boundary condition arising from science and engineering applications. We prove the well-posedness of the WG scheme by the monotone operator theory and the embedding inequality of weak finite element functions.
The error estimates are derived. Numerical experiments are presented to verify the
theoretical analysis. 相似文献
9.
Omar al-Khayat Are Magnus Bruaset & Hans Petter Langtangen 《Communications In Computational Physics》2011,10(4):823-843
This paper presents an extension of the lumped particle model in [1] to include
the effects of particle collisions. The lumped particle model is a flexible framework
for the modeling of particle laden flows, that takes into account fundamental
features, including advection, diffusion and dispersion of the particles. In this paper,
we transform a binary collision model and concepts from kinetic theory into a
collision procedure for the lumped particle framework. We apply this new collision
procedure to investigate numerically the role of particle collisions in the hindered settling
effect. The hindered settling effect is characterized by an increase in the effective
drag coefficient CD that influences each particle in the flow. This coefficient is given by
CD =(1−φ)−nC∗D, where φ is the volume fraction of particles, C∗Dis the drag coefficient
for a single particle, and n ≃ 4.67 for creeping flow. We obtain an approximation for
CD/C∗Dby calculating the effective work done by collisions, and comparing that to the
work done by the drag force. In our numerical experiments, we observe a minimal
value of n = 3.0. Moreover, by allowing high energy dissipation, an approximation
for the classical value for creeping flow, n = 4.7, is reproduced. We also obtain high
values for n, up to n=6.5, which is consistent with recent physical experiments on the
sedimentation of sand grains. 相似文献
10.
In this paper, we develop, analyze and test local discontinuous Galerkin
(LDG) methods for solving the Degasperis-Procesi equation which contains nonlinear
high order derivatives, and possibly discontinuous or sharp transition solutions. The
LDG method has the flexibility for arbitrary hand p adaptivity. We prove the L2stability
for general solutions. The proof of the total variation stability of the schemes
for the piecewise constant P0case is also given. The numerical simulation results for
different types of solutions of the nonlinear Degasperis-Procesi equation are provided
to illustrate the accuracy and capability of the LDG method. 相似文献
11.
Min Zhang Juan Cheng Weizhang Huang & Jianxian Qiu 《Communications In Computational Physics》2020,27(4):1140-1173
The radiative transfer equation models the interaction of radiation with scattering and absorbing media and has important applications in various fields in science
and engineering. It is an integro-differential equation involving time, frequency, space
and angular variables and contains an integral term in angular directions while being hyperbolic in space. The challenges for its numerical solution include the needs
to handle with its high dimensionality, the presence of the integral term, and the development of discontinuities and sharp layers in its solution along spatial directions.
Its numerical solution is studied in this paper using an adaptive moving mesh discontinuous Galerkin method for spatial discretization together with the discrete ordinate
method for angular discretization. The former employs a dynamic mesh adaptation
strategy based on moving mesh partial differential equations to improve computational accuracy and efficiency. Its mesh adaptation ability, accuracy, and efficiency are
demonstrated in a selection of one- and two-dimensional numerical examples. 相似文献
12.
For certain types of elliptic boundary control problems, the boundary element method has considerable advantage over the traditional finite element or finite difference methods because of the reduction of dimensionality in computations. In this paper we examine a variant of such boundary integral methods based on Cauchy integrals. The cost functional here contains only finitely many quadratic terms related to sensory data at those finite interior points. We see that the numerical efficiency of this approach hinges largely on the complexity of the inverse of a certain boundary integral operator. In the case of a circle, such an inverse is readily obtainable and entire computations require only a small effort to yield useful numerical information about the optimal control. Other general situations are also discussed. 相似文献
13.
We consider the solution of the Helmholtz equation −∆u(x)−n(x)2ω2u(x) = f(x), x = (x,y), in a domain Ω which is infinite in x and bounded in y. We assume that f(x) is supported in Ω0 := {x ∈ Ω |a− < x < a+} and that n(x) is x-periodic in Ω\Ω0. We show how to obtain exact boundary conditions on the vertical segments, Γ− := {x ∈ Ω |x = a−} and Γ+ := {x ∈ Ω |x = a+}, that will enable us to find the solution on Ω0 ∪Γ+ ∪Γ−. Then the solution can be extended in Ω in a straightforward manner from the values on Γ− and Γ+. The exact boundary conditions as well as the extension operators are computed by solving local problems on a single periodicity cell. 相似文献
14.
A Highly Scalable Boundary Integral Equation and Walk-on-Spheres (BIE-WOS) Method for the Laplace Equation with Dirichlet Data
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In this paper, we study a highly scalable communication-free parallel domain boundary decomposition algorithm for the Laplace equation based on a hybrid method combining boundary integral equations and walk-on-spheres (BIE-WOS)
method, which provides a numerical approximation of the Dirichlet-to-Neumann
(DtN) mapping for the Laplace equation. The BIE-WOS is a local method on the
boundary of the domain and does not require a structured mesh, and only needs a
covering of the domain boundary by patches and a local mesh for each patch for a local BIE. A new version of the BIE-WOS method with second kind integral equations is
introduced for better error controls. The effect of errors from the Feynman-Kac formula
based path integral WOS method on the overall accuracy of the BIE-WOS method is
analyzed for the BIEs, especially in the calculation of the right hand sides of the BIEs.
For the special case of flat patches, it is shown that the second kind integral equation
of BIE-WOS method can be simplified where the local BIE solutions can be given in
closed forms. A key advantage of the parallel BIE-WOS method is the absence of communications during the computation of the DtN mapping on individual patches of
the boundary, resulting in a complete independent computation using a large number
of cluster nodes. In addition, the BIE-WOS has an intrinsic capability of fault tolerance for exascale computations. The nearly linear scalability of the parallel BIE-WOS
method on a large-scale cluster with 6400 CPU cores is verified for computing the DtN
mapping of exterior Laplace problems with Dirichlet data for several domains. 相似文献
15.
Houde Han Leevan Ling & Tomoya Takeuchi 《Communications In Computational Physics》2011,9(4):878-896
Detecting corrosion by electrical field can be modeled by a Cauchy problem
of Laplace equation in annulus domain under the assumption that the thickness of the
pipe is relatively small compared with the radius of the pipe. The interior surface of the
pipe is inaccessible and the nondestructive detection is solely based on measurements
from the outer layer. The Cauchy problem for an elliptic equation is a typical ill-posed
problem whose solution does not depend continuously on the boundary data. In this
work, we assume that the measurements are available on the whole outer boundary on
an annulus domain. By imposing reasonable assumptions, the theoretical goal here is
to derive the stabilities of the Cauchy solutions and an energy regularization method.
Relationship between the proposed energy regularization method and the Tikhonov
regularization with Morozov principle is also given. A novel numerical algorithm is
proposed and numerical examples are given. 相似文献
16.
We developed a novel direct optimization method to solve distributed optimal control of viscous Burgers' equation over a finite-time horizon by minimizing the distance between the state function and a desired target state profile along with the energy of the control. Through a novel linearization strategy, well-conditioned integral reformulations, optimal Gegenbauer barycentric quadratures, and nodal discontinuous Galerkin discretizations, the method reduces such optimal control problems into finite-dimensional, nonlinear programming problems subject to linear algebraic system of equations and discrete mixed path inequality constraints that can be solved easily using standard optimization software. The proposed method produces “an auxiliary control function” that provides a useful model to explicitly define the optimal controller of the state variable. We present an error analysis of the semidiscretization and full discretization of the weak form of the reduced equality constraint system equations to demonstrate the exponential convergence of the method. The accuracy of the proposed method is examined using two numerical examples for various target state functions in the existence/absence of control bounds. The proposed method is exponentially convergent in both space and time, thus producing highly accurate approximations using a significantly small number of collocation points. 相似文献
17.
A High Frequency Boundary Element Method for Scattering by Convex Polygons with Impedance Boundary Conditions
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S. N. Chandler-Wilde S. Langdon & M. Mokgolele 《Communications In Computational Physics》2012,11(2):573-593
We consider scattering of a time harmonic incident plane wave by a convex polygon with piecewise constant impedance boundary conditions. Standard finite or boundary element methods require the number of degrees of freedom to grow at least linearly with respect to the frequency of the incident wave in order to maintain accuracy. Extending earlier work by Chandler-Wilde and Langdon for the sound soft problem, we propose a novel Galerkin boundary element method, with the approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh with smaller elements closer to the corners of the polygon. Theoretical analysis and numerical results suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency of the incident wave. 相似文献
18.
Fast Multipole Accelerated Boundary Integral Equation Method for Evaluating the Stress Field Associated with Dislocations in a Finite Medium
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Degang Zhao Jingfang Huang & Yang Xiang 《Communications In Computational Physics》2012,12(1):226-246
In this paper, we develop an efficient numerical method based on the boundary integral equation formulation and new version of fast multipole method to solve
the boundary value problem for the stress field associated with dislocations in a finite
medium. Numerical examples are presented to examine the influence from material
boundaries on dislocations. 相似文献
19.
On Time-Splitting Pseudospectral Discretization for Nonlinear Klein-Gordon Equation in Nonrelativistic Limit Regime
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Xuanchun Dong Zhiguo Xu & Xiaofei Zhao 《Communications In Computational Physics》2014,16(2):440-466
In this work, we are concerned with a time-splitting Fourier pseudospectral
(TSFP) discretization for the Klein-Gordon (KG) equation, involving a dimensionless
parameter ε∈(0,1]. In the nonrelativistic limit regime, the small ε produces high oscillations
in exact solutions with wavelength of O(ε2) in time. The key idea behind the
TSFP is to apply a time-splitting integrator to an equivalent first-order system in time,
with both the nonlinear and linear subproblems exactly integrable in time and, respectively,
Fourier frequency spaces. The method is fully explicit and time reversible.
Moreover, we establish rigorously the optimal error bounds of a second-order TSFP
for fixed ε = O(1), thanks to an observation that the scheme coincides with a type of
trigonometric integrator. As the second task, numerical studies are carried out, with
special efforts made to applying the TSFP in the nonrelativistic limit regime, which are
geared towards understanding its temporal resolution capacity and meshing strategy
for O(ε2)-oscillatory solutions when 0 < ε ≪ 1. It suggests that the method has uniform
spectral accuracy in space, and an asymptotic O(ε−2∆t2) temporal discretization
error bound (∆t refers to time step). On the other hand, the temporal error bounds for
most trigonometric integrators, such as the well-established Gautschi-type integrator
in [6], are O(ε−4∆t2). Thus, our method offers much better approximations than the
Gautschi-type integrator in the highly oscillatory regime. These results, either rigorous
or numerical, are valid for a splitting scheme applied to the classical relativistic NLS
reformulation as well. 相似文献
20.
Meng Zhao Pedro Anjos John Lowengrub Wenjun Ying & Shuwang Li 《Communications In Computational Physics》2023,33(2):399-424
We investigate the nonlinear dynamics of a moving interface in a Hele-Shaw
cell subject to an in-plane applied electric field. We develop a spectrally accurate numerical method for solving a coupled integral equation system. Although the stiffness
due to the high order spatial derivatives can be removed using a small scale decomposition technique, the long-time simulation is still expensive since the evolving velocity of the interface drops dramatically as the interface expands. We remove this
physically imposed stiffness by employing a rescaling scheme, which accelerates the
slow dynamics and reduces the computational cost. Our nonlinear results reveal that
positive currents restrain finger ramification and promote the overall stabilization of
patterns. On the other hand, negative currents make the interface more unstable and
lead to the formation of thin tail structures connecting the fingers and a small inner
region. When no fluid is injected, and a negative current is utilized, the interface tends
to approach the origin and break up into several drops. We investigate the temporal
evolution of the smallest distance between the interface and the origin and find that it
obeys an algebraic law $(t_∗−t)^b,$ where $t_∗$ is the estimated pinch-off time. 相似文献