共查询到20条相似文献,搜索用时 31 毫秒
1.
The paper presents a parallel direct solver for multi-physics problems. The solver is dedicated for solving problems resulting from adaptive finite element method computations. The concept of finite element is actually replaced by the concept of the node. The computational mesh consists of several nodes, related to element vertices, edges, faces and interiors. The ordering of unknowns in the solver is performed on the level of nodes. The concept of the node can be efficiently utilized in order to recognize unknowns that can be eliminated at a given node of the elimination tree. The solver is tested on the exemplary three-dimensional multi-physics problem involving the computations of the linear acoustics coupled with linear elasticity. The three-dimensional tetrahedral mesh generation and the solver algorithm are modeled by using graph grammar formalism. The execution time and the memory usage of the solver are compared with the MUMPS solver. 相似文献
2.
Two‐by‐two block matrices arise in various applications, such as in domain decomposition methods or when solving boundary value problems discretised by finite elements from the separation of the node set of the mesh into ‘fine’ and ‘coarse’ nodes. Matrices with such a structure, in saddle point form arise also in mixed variable finite element methods and in constrained optimisation problems. A general algebraic approach to construct, analyse and control the accuracy of preconditioners for matrices in two‐by‐two block form is presented. This includes both symmetric and nonsymmetric matrices, as well as indefinite matrices. The action of the preconditioners can involve element‐by‐element approximations and/or geometric or algebraic multigrid/multilevel methods. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
3.
Guo-Ping Liang & Ping Liang 《计算数学(英文版)》1990,8(4):363-370
We present a non-conforming domain decomposition technique for solving elliptic problems with the finite element method. Functions in the finite element space associated with this method may be discontinuous on the boundary of subdomains. The sizes of the finite meshes, the kinds of elements and the kinds of interpolation functions may be different in different subdomains. So, this method is more convenient and more efficient than the conforming domain decomposition method. We prove that the solution obtained by this method has the same convergence rate as by the conforming method, and both the condition number and the order of the capacitance matrix are much lower than those in the conforming case. 相似文献
4.
Composite Grid Finite Element Method: Implementation and Iterative Solution with Inexact Subproblems
This paper concerns the composite grid finite element (FE) method for solving boundary value problems in the cases which require local grid refinement for enhancing the approximating properties of the corresponding FE space. A special interest is given to iterative methods based on natural decomposition of the space of unknowns and to the implementation of both the composite grid FEM and the iterative procedures for its solution. The implementation is important for gaining all benefits of the described methods. We also discuss the case of inexact subproblems, which can frequently arise in the course of hierarchical modelling. 相似文献
5.
A parallel solver for the adaptive finite element analysis is presented. The primary aim of this work has been to establish
an efficient parallel computational procedure which requires only local computations to update the solution of the system
of equations arising from the finite element discretization after a local mesh-adaptation step. For this reason a set of algorithms
has been developed (two-level domain decomposition, recursive hierarchical mesh-refinement, selective solution-update of linear
systems of equations) which operate upon general and easily available partitioning, meshing and linear systems solving algorithms.
AMS subject classification 15A23, 65N50, 65N60 相似文献
6.
Tamal Pramanick 《计算数学(英文版)》2021,39(4):493-517
We study spatially semidiscrete and fully discrete two-scale composite finite element method for approximations of the nonlinear parabolic equations with homogeneous Dirich-let boundary conditions in a convex polygonal domain in the plane.This new class of finite elements,which is called composite finite elements,was first introduced by Hackbusch and Sauter[Numer.Math.,75(1997),pp.447-472]for the approximation of partial differential equations on domains with complicated geometry.The aim of this paper is to introduce an efficient numerical method which gives a lower dimensional approach for solving par-tial differential equations by domain discretization method.The composite finite element method introduces two-scale grid for discretization of the domain,the coarse-scale and the fine-scale grid with the degrees of freedom lies on the coarse-scale grid only.While the fine-scale grid is used to resolve the Dirichlet boundary condition,the dimension of the finite element space depends only on the coarse-scale grid.As a consequence,the resulting linear system will have a fewer number of unknowns.A continuous,piecewise linear composite finite element space is employed for the space discretization whereas the time discretization is based on both the backward Euler and the Crank-Nicolson methods.We have derived the error estimates in the L∞(L2)-norm for both semidiscrete and fully discrete schemes.Moreover,numerical simulations show that the proposed method is an efficient method to provide a good approximate solution. 相似文献
7.
Xiaojing Dong Yinnian He Hongbo Wei Yuhong Zhang 《Advances in Computational Mathematics》2018,44(4):1295-1319
Based on the partition of unity method (PUM), a local and parallel finite element method is designed and analyzed for solving the stationary incompressible magnetohydrodynamics (MHD). The key idea of the proposed algorithm is to first solve the nonlinear system on a coarse mesh, divide the globally fine grid correction into a series of locally linearized residual problems on some subdomains derived by a class of partition of unity, then compute the local subproblems in parallel, and obtain the globally continuous finite element solution by assembling all local solutions together by the partition of unity functions. The main feature of the new method is that the partition of unity provide a flexible and controllable framework for the domain decomposition. Finally, the efficiency of our theoretical analysis is tested by numerical experiments. 相似文献
8.
Summary.
The mortar element method is a
non conforming finite element method with
elements based on domain decomposition. For the Laplace equation,
it yields an ill conditioned linear system. For solving the linear system,
the so called preconditioned conjugate gradient method in
a subspace is used. Preconditioners are
proposed, and estimates on condition numbers
and arithmetical complexity are given.
Finally, numerical experiments are presented.
Received
June 22, 1994 / Revised version received February 6, 1995 相似文献
9.
Hongyi Yu 《Applied Numerical Mathematics》1999,30(4):475-491
We describe a method for solving parabolic partial differential equations (PDEs) using local refinement in time. Different time steps are used in different spatial regions based on a domain decomposition finite element method. Extrapolation methods based on either a linearly implicit mid-point rule or a linearly implicit Euler method are used to integrate in time. Extrapolation methods are a better fit than BDF methods in our context since local time stepping in different spatial regions precludes history information. Some linear and nonlinear examples demonstrate the effectiveness of the method. 相似文献
10.
It is well known that the ordering of the unknowns can have a significant effect on the convergence of a preconditioned iterative method and on its implementation on a parallel computer. To do so, we introduce a block red-black coloring to increase the degree of parallelism in the application of the blockILU preconditioner for solving sparse matrices, arising from convection-diffusion equations discretized using the finite difference scheme (five-point operator). We study the preconditioned PGMRES iterative method for solving these linear systems. 相似文献
11.
In this paper, the Burgers’ equation is transformed into the linear diffusion equation by using the Hopf–Cole transformation. The obtained linear diffusion equation is discretized in space by the local discontinuous Galerkin method. The temporal discretization is accomplished by the total variation diminishing Runge–Kutta method. Numerical solutions are compared with the exact solution and the numerical solutions obtained by Adomian’s decomposition method, finite difference method, B-spline finite element method and boundary element method. The results show that the local discontinuous Galerkin method is one of the most efficient methods for solving the Burgers’ equation. Even with small viscosity coefficient, it can get the satisfied solution. 相似文献
12.
In this paper a mixed method, which combines the finite element method and the differential quadrature element method (DQEM), is presented for solving the time dependent problems. In this study, the finite element method is first used to discretize the spatial domain. The DQEM is then employed as a step-by-step DQM in time domain to solve the resulting initial value problem. The resulting algebraic equations can be solved by either direct or iterative methods. Two general formulations using the DQM are also presented for solving a system of linear second-order ordinary differential equations in time. The application of the formulation is then shown by solving a sample moving load problem. Numerical results show that the present mixed method is very efficient and reliable. 相似文献
13.
对非定常线性化Navier-Stokes方程提出了非协调流线扩散有限元方法.用向后Euler格式离散时间,用流线扩散法处理扩散项带来的非稳定性.速度采用不连续的分片线性逼近,压力采用分片常数逼近.得到了离散解的存在唯一性以及在一定范数意义下离散解的稳定性和误差估计. 相似文献
14.
基于二重网格的定常Navier-Stokes方程的局部和并行有限元算法 总被引:1,自引:1,他引:0
对二维定常的不可压缩的Navier-Stokes方程的局部和并行算法进行了研究.给出的算法是多重网格和区域分解相结合的算法,它是基于两个有限元空间:粗网格上的函数空间和子区域的细网格上的函数空间.局部算法是在粗网格上求一个非线性问题,然后在细网格上求一个线性问题,并舍掉内部边界附近的误差相对较大的解.最后,基于局部算法,通过有重叠的区域分解而构造了并行算法,并且做了算法的误差分析,得到了比标准有限元方法更好的误差估计,也对算法做了数值试验,数值结果通过比较验证了本算法的高效性和合理性. 相似文献
15.
V. I. Vasil’ev M. V. Vasil’eva V. S. Gladkikh V. P. Ilin D. Ya. Nikiforov D. V. Perevozkin G. A. Prokop’ev 《Journal of Applied and Industrial Mathematics》2018,12(4):785-796
Under consideration are the numerical methods for simulation of a fluid flow in fractured porous media. The fractures are taken into account explicitly by using a discrete fracture model. The formulated single-phase filtering problem is approximated by an implicit finite element method on unstructured grids that resolve fractures at the grid level. The systems of linear algebraic equations (SLAE) are solved by the iterative methods of domain decomposition in the Krylov subspaces using the KRYLOVlibrary of parallel algorithms. The results of solving some model problem are presented. A study is conducted of the efficiency of the computational implementation for various values of contrast coefficients which significantly affect the condition number and the number of iterations required for convergence of the method. 相似文献
16.
不可压缩流动的数值模拟是计算流体力学的重要组成部分. 基于有限元离散方法, 本文设计了不可压缩Navier-Stokes (N-S)方程支配流的若干并行数值算法. 这些并行算法可归为两大类: 一类是基于两重网格离散方法, 首先在粗网格上求解非线性的N-S方程, 然后在细网格的子区域上并行求解线性化的残差方程, 以校正粗网格的解; 另一类是基于新型完全重叠型区域分解技巧, 每台处理器用一局部加密的全局多尺度网格计算所负责子区域的局部有限元解. 这些并行算法实现简单, 通信需求少, 具有良好的并行性能, 能获得与标准有限元方法相同收敛阶的有限元解. 理论分析和数值试验验证了并行算法的高效性 相似文献
17.
An implementation of the p‐version of the finite element method for solving two‐dimensional linear elliptic problems on a shared‐memory parallel computer
is analyzed. The idea is to partition the problem among the available processors and perform computations corresponding to
different elements in parallel. The parallelization is based on a domain decomposition technique using the Lagrange multipliers.
The numerical experiments carried out on the Sequent system indicate very high performance of the mixed finite element algorithm
in terms of attained speedups.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
18.
I. B. Yadykin 《Doklady Mathematics》2016,93(3):344-347
A new approach to solving discrete Lyapunov matrix algebraic equations is based on methods for spectral decomposition of their solutions. Assuming that all eigenvalues of the matrices on the left-hand side of the equation lie inside the unit disk, it is shown that the matrix of the solution to the equation can be calculated as a finite sum of matrix bilinear quadratic forms made up by products of Faddeev matrices obtained by decomposing the resolvents of the matrices of the Lyapunov equation. For a linear autonomous stochastic discrete dynamic system, analytical expressions are obtained for the decomposition of the asymptotic variance matrix of system’s states. 相似文献
19.
Bi-parameter incremental unknowns (IU) alternating directional implicit (ADI) iterative methods are proposed for solving elliptic problems. Condition numbers of the coefficient matrices for these iterative schemes are carefully estimated. Theoretical analysis shows that the condition numbers are reduced significantly by IU method, and the iterative sequences produced by the bi-parameter incremental unknowns ADI methods converge to the unique solution of the linear system if the two parameters belong to a given parameter region. Numerical examples are presented to illustrate the correctness of the theoretical analysis and the effectiveness of the bi-parameter incremental unknowns ADI methods. 相似文献
20.
Summary. We study the additive and multiplicative Schwarz domain decomposition methods for elliptic boundary value problem of order
2 r based on an appropriate spline space of smoothness . The finite element method reduces an elliptic boundary value problem to a linear system of equations. It is well known that
as the number of triangles in the underlying triangulation is increased, which is indispensable for increasing the accuracy
of the approximate solution, the size and condition number of the linear system increases. The Schwarz domain decomposition
methods will enable us to break the linear system into several linear subsystems of smaller size. We shall show in this paper
that the approximate solutions from the multiplicative Schwarz domain decomposition method converge to the exact solution
of the linear system geometrically. We also show that the additive Schwarz domain decomposition method yields a preconditioner
for the preconditioned conjugate gradient method. We tested these methods for the biharmonic equation with Dirichlet boundary
condition over an arbitrary polygonal domain using cubic spline functions over a quadrangulation of the given domain. The computer experiments agree with our theoretical results.
Received December 28, 1995 / Revised version received November 17, 1998 / Published online September 24, 1999 相似文献