一类抛物型方程组的特征修正区域分解有限元方法

在实际生产和科学研究中,有许多物理问题的数学模型为抛物型方程组问题,如可压缩核废料污染问题,地下水资源问题,杨青提出了差分格式和有限元格式,应用先验估计得到了最优的l^2和L^2模误差估计,江城顺等利用交替方向有限元方法得到了H^1模和L^2模误差估计．杨国强等采用显式可解的三层差分格式求解二维方程组得到了H^1模误差估计．     

Analysis of three-dimensional interior acoustic fields by using the localized method of fundamental solutions

The traditional method of fundamental solutions has a full interpolation matrix, and thus its solution is computationally expensive, especially for large-scale problems with complicated domains. In this paper, we make a first attempt to apply the localized method of fundamental solutions for analysis of 3D interior acoustic fields. The present method first divides the whole computational domain into some overlapping subdomains, and then expresses physical variables as linear combinations of the fundamental solution in each subdomain. Finally, the method forms a sparse and banded system matrix by satisfying both governing equations at interior nodes and boundary conditions at boundary nodes. We provide four numerical experiments to verify the accuracy and the stability of the method. Comparisons of numerical results and computational time are also made between the present method, the method of fundamental solutions, and the COMSOL software.     

Parameter selection by discrete mollification and the numerical solution of the inverse heat conduction problem

A procedure for the numerical solution of the one-dimensional inverse heat conduction problem, based on the computaion of the solution associated with a suitable filtered version of the noisy data by discrete mollification is presented and a parameter choice criterion, which automatically determines the radius of mollification as a function of the amount of noise in the data, is introduced. Several numerical examples of interest are also analyzed, showing the accuracy and stability properties of the method.     

Convergence and comparison analysis of some numerical Schwarz methods

Summary Domain decomposition methods are a natural means for solving partial differential equations on multi-processors. The spatial domain of the equation is expressed as a collection of overlapping subdomains and the solution of an associated equation is solved on each of these subdomains. The global solution is then obtained by piecing together the subsolutions in some manner. For elliptic equations, the global solution is obtained by iterating on the subdomains in a fashion that resembles the classical Schwarz alternating method. In this paper, we examine the convergence behavior of different subdomain iteration procedures as well as different subdomain approximations. For elliptic equations, it is shown that certain iterative procedures are equivalent to block Gauss-Siedel and Jacobi methods. Using different subdomain approximations, an inner-outer iterative procedure is defined.M-matrix analysis yields a comparison of different inner-outer iterations.Dedicated to the memory of Peter HenriciThis work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48     

Locally divergence-preserving schemes of higher order

Violation of the divergence constraint on the magnetic flux density in magnetohydrodynamical (MHD) simulations leads to stability problems. It is therefore of great importance to numerically respect this intrinsic constraint. Since the divergence preservation is a local phenomenon inherent in the MHD-system it is appealing to mimic this property numerically by a locally divergence-preserving scheme. A common numerical technique for simulation of the MHD-system of conservation laws is the finite volume (FV) method. In [SISC 26 2005 pp. 1166] a local procedure to redistribute the numerical fluxes in a FV-scheme so that a discrete divergence operator vanishes was presented. This procedure stabilizes the base scheme and respects the accuracy to the second order level. The present note describes a development of the above procedure that complies with the finite volume framework, preserves a fourth order discrete divergence operator locally and retains the accuracy of a generic semi-discrete finite volume scheme up to fourth order. The redistribution of the numerical magnetic field fluxes is formulated in a standard conservative setting, making it trivial to implement the divergence-preserving modification in an existing FV-scheme; see [JCP 227 2008 pp.3405] for the details. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Domain Decomposition Method Modified by Characteristic Finite Element Procedure for System of Parabolic Equations with Discontinuous Coefficients

In this work, system of parabolic equations with discontinuous coefficients is studied. The domain decomposition method modified by a characteristic finite element procedure is applied. A function is defined to approximate the fluxes on inner boundaries by using the solution at the previous level. Thus the parallelism is achieved. Convergence analysis and error estimate are also presented.     

数学物理中的总体和局部场GL方法

为了求解物理化学生物材料和金融中的微分方程,提出了一种总体(Global)和局部(Local)场方法.微分方程的求解区域可以是有限域,无限域,或具曲面边界的部分无限域.其无限域包括有限有界不均匀介质区域.其不均匀介质区域被分划为若干子区域之和.在这含非均匀介质的无限区域,将微分方程的解显式地表示为在若干非均匀介质子区域上和局部子曲面的积分的递归和.把正反算的非线性关系递归地显式化.在无限均匀区域,微分方程的解析解被称为初始总体场.微分方程解的总体场相继地被各个非均匀介质子区域的局部散射场所修正.这种修正过程是一个子域接着另个子域逐步相继地进行的.一旦所有非均匀介质子区域被散射扫描和有限步更新过程全部完成后,微分方程的解就获得了.称其为总体和局部场的方法,简称为GL方法.GL方法完全地不同于有限元及有限差方法,GL方法直接地逐子域地组装逆矩阵而获得解.GL方法无需求解大型矩阵方程,它克服了有限元大型矩阵解的困难.用有限元及有限差方法求解无限域上的微分方程时,人为边界及其上的吸收边界条件是必需的和困难的,人为边界上的吸收边界条件的不精确的反射会降低解的精确度和毁坏反算过程.GL方法又克服了有限元和有限差方法的人为边界的困难.GL方法既不需要任何人为边界又不需要任何吸收边界条件就可以子域接子域逐步精确地求解无限域上的微分方程.有限元和有限差方法都仅仅是数值的方法,GL方法将解析解和数值方法相容地结合起来.提出和证明了三角的格林函数积分方程公式.证明了当子域的直经趋于零时,波动方程的GL方法的数值解收敛于精确解.GL方法解波动方程的误差估计也获得了.求解椭圆型,抛物线型,双曲线型方程的GL模拟计算结果显示出我们的GL方法具有准确,快速,稳定的许多优点.GL方法可以是有网,无网和半网算法.GL方法可广泛应用在三维电磁场,三维弹塑性力学场,地震波场,声波场,流场,量子场等方面.上述三维电磁场等应用领域的GL方法的软件已经由作者研制和发展了。     

An Eulerian‐Lagrangian substructuring domain decomposition method for unsteady‐state advection‐diffusion equations

We develop an Eulerian‐Lagrangian substructuring domain decomposition method for the solution of unsteady‐state advection‐diffusion transport equations. This method reduces to an Eulerian‐Lagrangian scheme within each subdomain and to a type of Dirichlet‐Neumann algorithm at subdomain interfaces. The method generates accurate and stable solutions that are free of artifacts even if large time‐steps are used in the simulation. Numerical experiments are presented to show the strong potential of the method. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:565–583, 2001     

Finite-difference schemes for parabolic problems on graphs

We consider a reaction–diffusion parabolic problem on branched structures. The Hodgkin–Huxley reaction–diffusion equations are formulated on each edge of the graph. The problems are coupled by some conjugation conditions at branch points. It is important to note that two different types of the flux conservation equations are considered. The first one describes a conservation of the axial currents at branch points, and the second equation defines the conservation of the current flowing at the soma in neuron models. We study three different types of finite-difference schemes. The fully implicit scheme is based on the backward Euler algorithm. The stability and convergence of the discrete solution is proved in the maximum norm, and the analysis is done by using the maximum principle method. In order to decouple computations at each edge of the graph, we consider two modified schemes. In the predictor algorithm, the values of the solution at branch points are computed by using an explicit approximation of the conservation equations. The stability analysis is done using the maximum principle method. In the predictor–corrector method, in addition to the previous algorithm, the values of the solution at the branch points are recomputed by an implicit algorithm, when the discrete solution is obtained on each subdomain. The stability of this algorithm is investigated numerically. The results of computational experiments are presented.     

On the numerical solution of singular two‐point boundary value problems: A domain decomposition homotopy perturbation approach

This paper reports a modified homotopy perturbation algorithm, called the domain decomposition homotopy perturbation method (DDHPM), for solving two‐point singular boundary value problems arising in science and engineering. The essence of the approach is to split the domain of the problem into a number of nonoverlapping subdomains. In each subdomain, a method based on a combination of HPM and integral equation formalism is implemented. The boundary condition at the right endpoint of each inner subdomain is established before deriving an iterative scheme for the components of the solution series. The accuracy and efficiency of the DDHPM are demonstrated by 4 examples (2 nonlinear and 2 linear). In comparison with the traditional HPM, the proposed domain decomposition HPM is highly accurate.     

An adaptive NS/ES-FEM approach for 2D contact problems using triangular elements

An adaptive contact analysis approach is presented for 2D solid mechanics problems using only triangular elements and the subdomain parametric variational principle (SPVP). The present approach is implemented for the node-based smoothed FEM (or NS-FEM), the edge-based smoothed FEM (ES-FEM) and the standard FEM models with automatically adaptive refinement scheme. A modified Coulomb frictional contact model and its corresponding discrete equations are introduced. The global discretized system equations are then formulated in an incremental form with the aid of the basic boundary value equations for friction contact and the subdomain parametric variational principle. A simple adaptive refining scheme is presented, and the Voronoi vertices are taken as candidate points to become new nodes because of duality property between the Voronoi diagrams and Delaunay triangulation. The present adaptive approach can properly simulate variable behaviors of a contact interface such as bonding/debonding, contacting/departing, and sticking/slipping. Several examples are presented to numerically validate the proposed approach via the comparison with reference solutions obtained by ABAQUS^®, and to investigate the effects of the various parameters used in the computations on the response of the contact system. The numerical results have demonstrated that the present adaptive contact analysis approach using the ES-FEM has higher accuracy and convergence rate in the strain energy than that using FEM and NS-FEM. However, the latter two methods can provide the lower and upper bound solution for the system strain energy, respectively.     

Inverse source problem for a diffusion equation involving the fractional spectral Laplacian

In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill‐posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach.     

Coupling nonlinear Stokes and Darcy flow using mortar finite elements

We study a system composed of a nonlinear Stokes flow in one subdomain coupled with a nonlinear porous medium flow in another subdomain. Special attention is paid to the mathematical consequence of the shear-dependent fluid viscosity for the Stokes flow and the velocity-dependent effective viscosity for the Darcy flow. Motivated by the physical setting, we consider the case where only flow rates are specified on the inflow and outflow boundaries in both subdomains. We recast the coupled Stokes–Darcy system as a reduced matching problem on the interface using a mortar space approach. We prove a number of properties of the nonlinear interface operator associated with the reduced problem, which directly yield the existence, uniqueness and regularity of a variational solution to the system. We further propose and analyze a numerical algorithm based on mortar finite elements for the interface problem and conforming finite elements for the subdomain problems. Optimal a priori error estimates are established for the interface and subdomain problems, and a number of compatibility conditions for the finite element spaces used are discussed. Numerical simulations are presented to illustrate the algorithm and to compare two treatments of the defective boundary conditions.     

A mixed hybrid Mortar method for solving flow in discrete fracture networks

We consider flow in discrete fracture networks made of 2D domains in intersection and solved with a mixed hybrid finite element method (MHFEM). The discretization within each fracture is performed in two steps: first, borders and intersections are discretized, second, based on these discretizations, a 2D mesh is built. Independent meshing process within each subdomain is of interest for practical use since it makes it easier to refine the chosen subdomains and to perform parallel computation. This article shows how MHFEM is well adapted for integrating a Mortar method to enforce the continuity of the fluxes and heads at the non-matching grids. Some numerical simulations are given to show the efficiency of the method in the case of a preferential orientation of the fractures where a comparison with the 2D solution is possible.     

Hierarchical-matrix method for a class of diffusion-dominated partial integro-differential equations

A hierarchical matrix approach for solving diffusion-dominated partial integro-differential problems is presented. The corresponding diffusion-dominated differential operator is discretized by a second-order accurate finite-volume scheme, while the Fredholm integral term is approximated by the trapezoidal rule. The hierarchical matrix approach is used to approximate the resulting algebraic problem and includes the implementation of an efficient preconditioned generalized minimum residue (GMRes) solver. This approach extends previous work on integral forms of boundary element methods by taking into account inherent characteristics of the diffusion-dominated differential operator in the resultant algebraic problem. Numerical analysis estimates of the accuracy and stability of the finite-volume and the trapezoidal rule approximation are presented and combined with estimates of the hierarchical-matrix approximation and with the accuracy of the GMRes iterates. Results of numerical experiments are reported that successfully validate the theoretical accuracy and convergence estimates, and demonstrate the almost optimal computational complexity of the proposed solution procedure.     

A Finite Difference Scheme on a Priori Adapted Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Equation

A boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation;we construct a finite difference scheme on a priori (se-quentially) adapted meshes and study its convergence.The scheme on a priori adapted meshes is constructed using a majorant function for the singular component of the discrete solution,which allows us to find a priori a subdomain where the computed solution requires a further improvement.This subdomain is defined by the perturbation parameterε,the step-size of a uniform mesh in x,and also by the required accuracy of the discrete solution and the prescribed number of refinement iterations K for im- proving the solution.To solve the discrete problems aimed at the improvement of the solution,we use uniform meshes on the subdomains.The error of the numerical so- lution depends weakly on the parameterε.The scheme converges almostε-uniformly, precisely,under the condition N~(-1)=o(ε~v),where N denotes the number of nodes in the spatial mesh,and the value v=v(K) can be chosen arbitrarily small for suitable K.     

Central ADER schemes for hyperbolic conservation laws

In this paper we first briefly review the very high order ADER methods for solving hyperbolic conservation laws. ADER methods use high order polynomial reconstruction of the solution and upwind fluxes as the building block. They use a first order upwind Godunov and the upwind second order weighted average (WAF) fluxes. As well known the upwind methods are more accurate than central schemes. However, the superior accuracy of the ADER upwind schemes comes at a cost, one must solve exactly or approximately the Riemann problems (RP). Conventional Riemann solvers are usually complex and are not available for many hyperbolic problems of practical interest. In this paper we propose to use two central fluxes, instead of upwind fluxes, as the building block in ADER scheme. These are the monotone first order Lax-Friedrich (LXF) and the third order TVD flux. The resulting schemes are called central ADER schemes. Accuracy of the new schemes is established. Numerical implementations of the new schemes are carried out on the scalar conservation laws with a linear flux, nonlinear convex flux and non-convex flux. The results demonstrate that the proposed scheme, with LXF flux, is comparable to those using first and second order upwind fluxes while the scheme, with third order TVD flux, is superior to those using upwind fluxes. When compared with the state of art ADER schemes, our central ADER schemes are faster, more accurate, Riemann solver free, very simple to implement and need less computer memory. A way to extend these schemes to general systems of nonlinear hyperbolic conservation laws in one and two dimensions is presented.     

Mixed finite volume methods for semiconductor device simulation

We deal with the two-dimensional numerical solution of the Van Roosbroeck system, widely employed in modern semiconductor device simulation. Using the well-known Gummel's decoupled algorithm leads to the iterative solution of a nonlinear Poisson equation for the electric potential and two linearized continuity equations for the electron and hole current densities. The numerical approximation is based on the dual mixed formulation for a self-adjoint second-order elliptic operator by using the Raviart-Thomas (RT) finite elements of lowest degree on a triangular partition of the device domain. In this article, we propose a suitable variant of the RT method, based on the diagonalization of the element mass matrix. This is achieved by use of an appropriate numerical integration that eliminates the fluxes and gives rise to a cell-centered finite volume scheme for the scalar unknown with the same approximation properties of the mixed approach, but at a reduced computational cost. The above procedure suggests also a natural way to introduce in the frame of the classical Box Method (BM) suitable vector basis functions (edge elements) to represent the current field over each mesh triangle. This issue may be profitably employed both as a postprocessing tool, as well as a technique for solving the current continuity equations when source terms depending on the current itself are included in the mathematical model. Simulations of realistic semiconductor devices are then included to demonstrate the accuracy and stability of the new method. © 1997 John Wiley & Sons, Inc.     

一类无结构三角网上抛物方程的有限差分区域分解算法

本文讨论了一类在无结构三角网上数值求解二维热传导方程的有限差分区域分解算法．在这个算法中,将通过引进两类不同类型的内界点,将求解区域分裂成若干子区域．一旦内界点处的值被计算出来,其余子区域上的计算可完全并行．本文得到了稳定性条件和最大模误差估计,它表明我们的格式有令人满意的稳定性和较高的收敛阶．     

On parallel asynchronous high-order solutions of parabolic PDEs

In achieving significant speed-up on parallel machines, a major obstacle is the overhead associated with synchronizing the concurrent processes. This paper presents high-orderparallel asynchronous schemes, which are schemes that are specifically designed to minimize the associated synchronization overhead of a parallel machine in solving parabolic PDEs. They are asynchronous in the sense that each processor is allowed to advance at its own speed. Thus, these schemes are suitable for single (or multi) user shared memory or (message passing) MIMD multiprocessors. Our approach is demonstrated for the solution of the multidimensional heat equation, of which we present a spatial second-order Parametric Asynchronous Finite-Difference (PAFD) scheme. The well-known synchronous schemes are obtained as its special cases. This is a generalization and expansion of the results in [5] and [7]. The consistency, stability and convergence of this scheme are investigated in detail. Numerical tests show that although PAFD provides the desired order of accuracy, its efficiency is inadequate when performed on each grid point.In an alternative approach that uses domain decomposition, the problem domain is divided among the processors. Each processor computes its subdomain mostly independently, while the PAFD scheme provides the solutions at the subdomains' boundaries. We use high-order finite-difference implicit scheme within each subdomain and determine the values at subdomains' boundaries by the PAFD scheme. Moreover, in order to allow larger time-step, we use remote neighbors' values rather than those of the immediate neighbors. Numerical tests show that this approach provides high efficiency and in the case which uses remote neighbors' values an almost linear speedup is achieved. Schemes similar to the PAFD can be developed for other types of equations [3].This research was supported by the fund for promotion of research at the Technion.