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1.
Calculating the vertex unknowns of nine point scheme on quadrilateral meshes for diffusion equation 总被引:1,自引:0,他引:1
In the construction of nine point scheme,both vertex unknowns and cell-centered unknowns are introduced,and the vertex unknowns are usually eliminated by using the interpolation of neighboring cell-centered unknowns,which often leads to lose accuracy.Instead of using interpolation,here we propose a different method of calculating the vertex unknowns of nine point scheme,which are solved independently on a new generated mesh.This new mesh is a Vorono¨i mesh based on the vertexes of primary mesh and some additional points on the interface.The advantage of this method is that it is particularly suitable for solving diffusion problems with discontinuous coeffcients on highly distorted meshes,and it leads to a symmetric positive definite matrix.We prove that the method has first-order convergence on distorted meshes.Numerical experiments show that the method obtains nearly second-order accuracy on distorted meshes. 相似文献
2.
Yubo Zhang & Tao Tang 《高等学校计算数学学报(英文版)》2011,4(1):92-112
An efficient finite difference framework based on moving meshes
methods is developed for the three-dimensional free surface
viscoelastic flows. The basic model equations are based on the
incompressible Navier-Stokes equations and the Oldroyd-B
constitutive model for viscoelastic flows is adopted. A logical
domain semi-Lagrangian scheme is designed for moving-mesh solution
interpolation and convection. Numerical results show that harmonic
map based moving mesh methods can achieve better accuracy for
viscoelastic flows with free boundaries while using much less memory
and computational time compared to the uniform mesh simulations. 相似文献
3.
A. I. Zadorin 《Computational Mathematics and Mathematical Physics》2008,48(9):1634-1645
Linear and quadratic spline interpolation methods for a one-variable function with a boundary-layer component are examined. It is shown that the interpolation method for such a function leads to considerable errors when applied on a uniform mesh. The error of linear and quadratic spline interpolations on meshes that are refined in the boundary layer is estimated. Numerical results are presented. 相似文献
4.
A singularly perturbed one-dimensional convection-diffusion problem is solved numerically by the finite element method based on higher order polynomials. Numerical solutions are obtained using S-type meshes with special emphasis on meshes which are graded (based on a mesh generating function) in the fine mesh region. Error estimates in the ε-weighted energy norm are proved. We derive an 'optimal' mesh generating function in order to minimize the constant in the error estimate. Two layer-adapted meshes defined by a recursive formulae in the fine mesh region are also considered and a new technique for proving error estimates for these meshes is presented. The aim of the paper is to emphasize the importance of using optimal meshes for higher order finite element methods. Numerical experiments support all theoretical results. 相似文献
5.
A superconvergent fitted finite volume method for Black–Scholes equations governing European and American option valuation
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Song Wang Shuhua Zhang Zhiwei Fang 《Numerical Methods for Partial Differential Equations》2015,31(4):1190-1208
We develop a superconvergent fitted finite volume method for a degenerate nonlinear penalized Black–Scholes equation arising in the valuation of European and American options, based on the fitting idea in Wang [IMA J Numer Anal 24 (2004), 699–720]. Unlike conventional finite volume methods in which the dual mesh points are naively chosen to be the midpoints of the subintervals of the primal mesh, we construct the dual mesh judiciously using an error representation for the flux interpolation so that both the approximate flux and solution have the second‐order accuracy at the mesh points without any increase in computational costs. As the equation is degenerate, we also show that it is essential to refine the meshes locally near the degenerate point in order to maintain the second‐order accuracy. Numerical results for both European and American options with constant and nonconstant coefficients will be presented to demonstrate the superconvergence of the method. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1190–1208, 2015 相似文献
6.
In our paper we present a method to the computational simulation of flow problems on domains with translating boundaries. Our moving mesh algorithm, defined on special tetrahedral meshes, avoids global interpolation and re‐meshing, thus it works quite efficiently on problems with strongly deforming domains while ensures conservativity. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
7.
In this paper, we propose three gradient recovery schemes of higher order for the linear interpolation. The first one is a weighted averaging method based on the gradients of the linear interpolation on the uniform mesh, the second is a geometric averaging method constructed from the gradients of two cubic interpolation on macro element, and the last one is a local least square method on the nodal patch with cubic polynomials. We prove that these schemes can approximate the gradient of the exact solution on the symmetry points with fourth order. In particular, for the uniform mesh, we show that these three schemes are the same on the considered points. The last scheme is more robust in general meshes. Consequently, we obtain the superconvergence results of the recovered gradient by using the aforementioned results and the supercloseness between the finite element solution and the linear interpolation of the exact solution. Finally, we provide several numerical experiments to illustrate the theoretical results. 相似文献
8.
J. Southern G.J. Gorman M.D. Piggott P.E. Farrell 《Journal of computational science》2012,3(1-2):8-16
Simulations in cardiac electrophysiology generally use very fine meshes and small time steps to resolve highly localized wavefronts. This expense motivates the use of mesh adaptivity, which has been demonstrated to reduce the overall computational load. However, even with mesh adaptivity performing such simulations on a single processor is infeasible. Therefore, the adaptivity algorithm must be parallelised. Rather than modifying the sequential adaptive algorithm, the parallel mesh adaptivity method introduced in this paper focuses on dynamic load balancing in response to the local refinement and coarsening of the mesh. In essence, the mesh partition boundary is perturbed away from mesh regions of high relative error, while also balancing the computational load across processes. The parallel scaling of the method when applied to physiologically realistic heart meshes is shown to be good as long as there are enough mesh nodes to distribute over the available parallel processes. It is shown that the new method is dominated by the cost of the sequential adaptive mesh procedure and that the parallel overhead of inter-process data migration represents only a small fraction of the overall cost. 相似文献
9.
Adaptive computation using adaptive meshes is now recognized as essential for solving complex PDE problems. This computation requires, at each step, the definition of a continuous metric field to govern the generation of the adapted meshes. In practice, via an appropriate a posteriori error estimation, metrics are calculated at the vertices of the computational domain mesh. In order to obtain a continuous metric field, the discrete field is interpolated in the whole domain mesh. In this Note, a new method for interpolating discrete metric fields, based on a so-called “natural decomposition” of metrics, is introduced. The proposed method is based on known matrix decompositions and is computationally robust and efficient. Some qualitative comparisons with classical methods are made to show the relevance of this methodology. 相似文献
10.
A new numerical method based on locally modified Cartesian meshes is proposed for solving a coupled system of a fluid flow and a porous media flow. The fluid flow is modeled by the Stokes equations while the porous media flow is modeled by Darcy's law. The method is based on a Robin-Robin domain decomposition method with a Cartesian mesh with local modifications near the interface. Some computational examples are presented and discussed. 相似文献
11.
1.引言设0是RZ中的有界多边形区域,其边界为Rfl.考虑下面的重调和Dirichlet问题:(1.1)的变分形式为:求。EHI(fi)使得对?/EL‘(m,问题(1.幻的唯一可解性可由冯(m上的M线性型的强制性和连续性以及La。Mlgram定理得出(of[4]).令人一{丸)是n的一个三角剖分,并且满足最小角条件,其中h是它的网格参数.设Vh为Money元空间[41.问题(1.2)的有限元离散问题为:求。eVh使得当有限元参数人很小时,这个方程组很大,而且矩阵A的条件数变得非常大,直接求解,存贮量及计算量都很大.如果B可逆,则方程组(1.4)等… 相似文献
12.
Csaba Gspr 《PAMM》2004,4(1):640-641
Meshless methods have become quite popular in numerical treatment of partial differential equations because of their simplicity and the fact that they require neither domain nor boundary mesh. In general, however, they convert the original problem to a highly ill‐conditioned linear system of algebraic equations with a dense matrix. Recently, a special technique has been proposed which circumvents this computational difficulty. This method, called Direct Multi‐Elliptic Interpolation Method, is based on a scattered data interpolation which defines the interpolation function as a solution of a higher order multi‐elliptic equation. Here the boundary version of this meshless method which is based on a multi‐elliptic boundary interpolation is considered. Error estimations are derived justifying the interpolation function to be a good approximation of the solution of the original boundary value problem as well. At the same time, the problem of large, dense and ill‐conditioned matrices as well as the mesh generation are completely avoided. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
13.
The quality of a finite element solution has been shown to be affected by the quality of the underlying mesh. A poor mesh may lead to unstable and/or inaccurate finite element approximations. Mesh quality is often characterized by the “smoothness” or “shape” of the elements (triangles in 2-D or tetrahedra in 3-D). Most automatic mesh generators produce an initial mesh where the aspect ratio of the elements are unacceptably high. In this paper, a new approach to produce acceptable quality meshes from a topologically valid initial mesh is presented. Given an initial mesh (nodal coordinates and element connectivity), a “smooth” final mesh is obtained by solving a constrained optimization problem. The variables for the iterative optimization procedure are the nodal coordinates (excluding, the boundary nodes) of the finite element mesh, and appropriate bounds are imposed on these to prevent an unacceptable finite element mesh. Examples are given of the application of the above method for 2- and 3-D meshes generated using
automatic mesh generators. Results indicate that the new method not only yields better quality elements when compared with the traditional Laplacian smoothing, but also guarantees a valid mesh unlike the Laplacian method. 相似文献
14.
本文考虑求解Helmholtz方程的有限元方法的超逼近性质以及基于PPR后处理方法的超收敛性质.我们首先给出了矩形网格上的p-次元在收敛条件k(kh)2p+1≤C0下的有限元解和基于Lobatto点的有限元插值之间的超逼近以及重构的有限元梯度和精确解之间的超收敛分析.然后我们给出了四边形网格上的线性有限元方法的分析.这些估计都给出了与波数k和网格尺寸h的依赖关系.同时我们回顾了三角形网格上的线性有限元的超收敛结果.最后我们给出了数值实验并且结合Richardson外推进一步减少了误差. 相似文献
15.
We present a new algorithm for generating layer-adapted meshes for the
finite element solution of singularly perturbed problems based on mesh partial differential equations (MPDEs). The ultimate goal is to design meshes that are similar
to the well-known Bakhvalov meshes, but can be used in more general settings:
specifically two-dimensional problems for which the optimal mesh is not tensor-product in nature. Our focus is on the efficient implementation of these algorithms,
and numerical verification of their properties in a variety of settings. The MPDE is
a nonlinear problem, and the efficiency with which it can be solved depends adversely on the magnitude of the perturbation parameter and the number of mesh
intervals. We resolve this by proposing a scheme based on $h$-refinement. We present
fully working FEniCS codes [Alnaes et al., Arch. Numer. Softw., 3 (100) (2015)]
that implement these methods, facilitating their extension to other problems and
settings. 相似文献
16.
Xiangyu Shi & Linzhang Lu 《计算数学(英文版)》2023,41(1):94-106
This article aims to study the unconditional superconvergent behavior of nonconforming quadrilateral quasi-Wilson element for nonlinear Benjamin Bona Mahoney (BBM) equation. For the generalized rectangular meshes including rectangular mesh, deformed rectangular mesh and piecewise deformed rectangular mesh, by use of the special character of this element, that is, the conforming part (bilinear element) has high accuracy estimates on the generalized rectangular meshes and the consistency error can reach order $O(h^2)$, one order higher than its interpolation error, the superconvergent estimates with respect to mesh size $h$ are obtained in the broken $H^1$-norm for the semi-/ fully-discrete schemes. A striking ingredient is that the restrictions between mesh size $h$ and time step $\tau$ required in the previous works are removed. Finally, some numerical results are provided to confirm the theoretical analysis. 相似文献
17.
F. Piazzon 《Applicable analysis》2013,92(5):1063-1073
We show that the property of being a (weakly) admissible mesh for multivariate polynomials is preserved by small perturbations on real and complex Markov compacts. Applications are given to smooth transformations of polynomial meshes and to polynomial interpolation. 相似文献
18.
This paper is concerned with a specific finite element strategy for solving elliptic boundary value problems in domains with corners and edges. First, the anisotropic singular behaviour of the solution is described. Then the finite element method with anisotropic, graded meshes and piecewise linear shape functions is investigated for such problems; the schemes exhibit optimal convergence rates with decreasing mesh size. For the proof, new local interpolation error estimates for functions from anisotropically weighted spaces are derived. Finally, a numerical experiment is described, that shows a good agreement of the calculated approximation orders with the theoretically predicted ones. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd. 相似文献
19.
M. V. Muratov I. B. Petrov I. V. Sannikov A. V. Favorskaya 《Computational Mathematics and Mathematical Physics》2014,54(5):837-847
The goal of this paper is to develop a grid-characteristic method intended for high-performance computer systems and implemented on unstructured tetrahedral hierarchical meshes with the use of a multiple time step and high-order interpolation, including interpolation with a limiter, piecewise parabolic interpolation, and monotone interpolation. The method is designed for simulating complex three-dimensional dynamical processes in heterogeneous media. It involves accurately stated contact conditions and produces physically correct solutions of problems in seismology and seismic exploration. Hierarchical meshes make it possible to take into account numerous inhomogeneous inclusions (cracks, cavities, etc.) and to solve problems in a real-life formulation. The grid-characteristic method enables the use of a multiple time step. As a result, the computation time is considerably reduced and the efficiency of the method is raised. The method is parallelized on a computer cluster with an optimal use of system resources. 相似文献
20.
We survey multilevel iterative methods applied for solving large sparse systems with matrices, which depend on a level parameter, such as arise by the discretization of boundary value problems for partial differential equations when successive refinements of an initial discretization mesh is used to construct a sequence of nested difference or finite element meshes.We discuss various two-level (two-grid) preconditioning techniques, including some for nonsymmetric problems. The generalization of these techniques to the multilevel case is a nontrivial task. We emphasize several ways this can be done including classical multigrid methods and a recently proposed algebraic multilevel preconditioning method. Conditions for which the methods have an optimal order of computational complexity are presented.On leave from the Institute of Mathematics, and Center for Informatics and Computer Technology, Bulgarian Academy of Sciences, Sofia, Bulgaria. The research of the second author reported here was partly supported by the Stichting Mathematisch Centrum, Amsterdam. 相似文献