两类非线性双曲型方程组有限元方法的误差估计

对两类非线性双曲型方程组给出了全离散有限元逼近格式,并得到最优H^1模和L^2模误差估计。     

非线性双曲型积分微分方程有限元逼近的误差分析

考虑非线性双曲型积分微分方程半离散有限元格式，得到H^1超收敛和最优阶L^∞和W^1,∞模误差估计，结果丰富了有限元方法的理论。     

水污染问题特征有限元方法的数值计算及理论分析

本文研究了水污染二维对流占优数学模型特征有限元方法的计算问题，导出的计算格式对时间变量用特征线方法离散，对空间变量用Galerkin有限元方法离散，得到的H^1－模和L^2－模误差估计是最优阶的。     

一类反应扩散方程组的隐-显多步有限元方法及其分析

考虑一类非线性反应扩散方程组，提出了隐-显多步有限元格式逼近，证明了格式最优的L^2模误差估计．     

美式回望期权定价的有限元超收敛分析

考虑美式回望看跌期权的有限元方法．在把原问题转化成等价的变分不等式的基础上,研究了半离散格式在L^2和L^∞范数意义下的最优误差估计．此外,为了进一步提高逼近解的精度,借助超收敛分析技术和插值后处理方法,研究了H^1范数意义下的整体超收敛以及后验误差估计。     

三维热传导型半导体器件瞬态模拟问题的Crank—Nicolson差分—流线扩散有限元法及其数值分析

本文研究三维热传导型半导体器件瞬态模拟问题的数值方法。针对数学模型中各方程不同的特点，分别提出不同的有限元格式。特别针对浓度方程组是对流为主扩散问题的特点，使用Crank-Nicolson差分-流线扩散计算格式，提高了数值解的稳定性。得到的L^2误差估计关于空间剖分步长是拟最优的，关于时间步长具有二阶精度。     

一类非线性抛物型方程组的交替方向变网格有限元方法

本文对一类非线性抛物型方程组提出并分析了一类全离散交替方向变网格有限元格式，且在相当一般的情况下得到了最佳的L^2模误差估计。     

荷载属于H^—1（Ω）时不完全双次非协调板元的新的误差估计式

在真解u∈H^3(Ω），荷载f∈H^-1(Ω）时，通过修正的离散格式，给出了不完全双次非协调板元的能量模及L^2-模新的误差估计式，它改善了以往相应的结果且减少了计算工作量。     

反应扩散方程的紧交替方向差分格式

本文研究二维常系数反应扩散方程的紧交替方向隐式差分格式．首先综合应用降阶法和降维法导出了紧差分格式,并给出了差分格式截断误差的表达式．其次引进过渡层变量,给出了紧交替方向隐式差分格式算法．接着用能量分析方法给出了紧交替方向隐式差分格式的解在离散H^1范数下的先验估计式,证明了差分格式的可解性、稳定性和收敛性,在离散H^1范数下收敛阶为O(r^2 H^4)．然后将Rechardson外推法应用于紧交替方向隐式差分格式,外推一次得到具有O(r^4 H^6)阶精度的近似解．最后给出了数值例子,数值结果和理论结果是吻合的．     

二维三温热传导方程组的分数步隐式差分格式

本文给出一个解二维三温热传导方程组的分数步隐式有限差分格式，利用离散变分形式及能量方法，给出差分格式的最优阶离散H^1范数先验误差及稳定性估计。     

A fast numerical method for solving coupled Burgers' equations

A new fast numerical scheme is proposed for solving time‐dependent coupled Burgers' equations. The idea of operator splitting is used to decompose the original problem into nonlinear pure convection subproblems and diffusion subproblems at each time step. Using Taylor's expansion, the nonlinearity in convection subproblems is explicitly treated by resolving a linear convection system with artificial inflow boundary conditions that can be independently solved. A multistep technique is proposed to rescue the possible instability caused by the explicit treatment of the convection system. Meanwhile, the diffusion subproblems are always self‐adjoint and coercive at each time step, and they can be efficiently solved by some existing preconditioned iterative solvers like the preconditioned conjugate galerkin method, and so forth. With the help of finite element discretization, all the major stiffness matrices remain invariant during the time marching process, which makes the present approach extremely fast for the time‐dependent nonlinear problems. Finally, several numerical examples are performed to verify the stability, convergence and performance of the new method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1823–1838, 2017     

Stabilized finite element schemes for generalized Oseen systems

In recent works [1, 2], advanced approximation schemes for the numerical calculation of compressible viscous flow were developed, analyzed theoretically and applied successfully to benchmark problems. These methods are based on splitting the Poisson–Stokes system (1.1) describing the motion of a viscous compressible fluid into a generalized Oseen problem for the velocity and a hyperbolic transport equation for the density. Highly refined finite element techniques were proposed for the numerical solution of these separated subproblems of simpler structure; cf. [2]. In this paper, error estimates for a SUPG/(PSPG) and grad-div stabilized finite element approximation of the generalized Oseen problems that arise in the course of the splitting procedure are presented. LBB-stable pairs of finite element spaces are used for velocity and pressure. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A coupling of FEM-BEM for a kind of Signorini contact problem

In this paper, we consider a kind of coupled nonlinear problem with Signorini contact conditions. To solve this problem, we discuss a new coupling of finite element and boundary element by adding an auxiliary circle. We first derive an asymptotic error estimate of the approximation to the coupled FEM-BEM variational inequality. Then we design an iterative method for solving the coupled system, in which only three standard subproblems without involving any boundary integral equation are solved. It will be shown that the convergence speed of this iteration method is independent of the mesh size.     

A finite element — Capacitance method for elliptic problems on regions partitioned into subregions

Summary A method is given for the solution of linear equations arising in the finite element method applied to a general elliptic problem. This method reduces the original problem to several subproblems (of the same form) considered on subregions, and an auxiliary problem. Very efficient iterative methods with the preconditioning operator and using FFT are developed for the auxiliary problem.     

An element agglomeration nonlinear additive Schwarz preconditioned Newton method for unstructured finite element problems

This paper extends previous results on nonlinear Schwarz preconditioning (Cai and Keyes 2002) to unstructured finite element elliptic problems exploiting now nonlocal (but small) subspaces. The nonlocal finite element subspaces are associated with subdomains obtained from a non-overlapping element partitioning of the original set of elements and are coarse outside the prescribed element subdomain. The coarsening is based on a modification of the agglomeration based AMGe method proposed in Jones and Vassilevski 2001. Then, the algebraic construction from Jones, Vassilevski and Woodward 2003 of the corresponding non-linear finite element subproblems is applied to generate the subspace based nonlinear preconditioner. The overall nonlinearly preconditioned problem is solved by an inexact Newton method. A numerical illustration is also provided.This work was performed under the auspices of the U.S. Department of Energy by the University of California Lawrence Livermore National Laboratory: contract/grant number: W-7405-Eng-48. The contribution of the second author was also partially supported by Polish Scientific Grant 2/P03A/005/24.     

A non-overlapping domain decomposition dual reciprocity method for solving the forward–backward heat equation in two-dimension

We apply a boundary element dual reciprocity method (DRBEM) to the numerical solution of the forward–backward heat equation in a two-dimensional case. The method is employed for the spatial variable via the fundamental solution of the Laplace equation and the Crank–Nicolson finite difference scheme is utilized to treat the time variable. The physical domain is divided into two non-overlapping subdomains resulting in two standard forward and backward parabolic equations. The subproblems are then treated by the underlying method assuming a virtual boundary in the interface and starting with an initial approximate solution on this boundary followed by updating the solution by an iterative procedure. In addition, we show that the time discrete scheme is unconditionally stable and convergent using the energy method. Furthermore, some computational aspects will be suggested to efficiently deal with the formulation of the proposed method. Finally, two forward–backward problems, for which the exact solution is available, will be numerically solved for two different domains to demonstrate the efficiency of the proposed approach.     

Analytical solution for the evolution of exit point along the sloping seepage face

This paper developed an analytical solution for the problem of exit point evolution on the seepage face in the unconfined aquifer with sloping interface. A theoretical model for the groundwater drawdown problem in a half‐infinite aquifer with a sloping boundary is built in accordance with the linearized one‐dimensional Boussinesq equation and the Neumann boundary condition at the seepage point. The homotopy analysis method is then adopted for solving this dynamic boundary problem. By constructing two continuous deformations, the original problem could be converted into a group of subproblems with the same physical essence and similar mathematical solutions. To compare this analytical solution, a numerical model based on the finite volume method is developed, which employs adaptive grids to settle the dynamic boundary condition. The comparisons show that the analytical solution agrees with the numerical model well. The results are useful for the quantification of various hydrological problems. The methodology applied in this study is referential for other dynamic boundary problems as well.     

Numerical electroseismic modeling: A finite element approach

Electroseismics is a procedure that uses the conversion of electromagnetic to seismic waves in a fluid-saturated porous rock due to the electrokinetic phenomenon. This work presents a collection of continuous and discrete time finite element procedures for electroseismic modeling in poroelastic fluid-saturated media. The model involves the simultaneous solution of Biot’s equations of motion and Maxwell’s equations in a bounded domain, coupled via an electrokinetic coefficient, with appropriate initial conditions and employing absorbing boundary conditions at the artificial boundaries. The 3D case is formulated and analyzed in detail including results on the existence and uniqueness of the solution of the initial boundary value problem. Apriori error estimates for a continuous-time finite element procedure based on parallelepiped elements are derived, with Maxwell’s equations discretized in space using the lowest order mixed finite element spaces of Nédélec, while for Biot’s equations a nonconforming element for each component of the solid displacement vector and the vector part of the Raviart-Thomas-Nédélec of zero order for the fluid displacement vector are employed. A fully implicit discrete-time finite element method is also defined and its stability is demonstrated. The results are also extended to the case of tetrahedral elements. The 2D cases of compressional and vertically polarized shear waves coupled with the transverse magnetic polarization (PSVTM-mode) and horizontally polarized shear waves coupled with the transverse electric polarization (SHTE-mode) are also formulated and the corresponding finite element spaces are defined. The 1D SHTE initial boundary value problem is also formulated and approximately solved using a discrete-time finite element procedure, which was implemented to obtain the numerical examples presented.     

On the coupling of BEM and FEM for exterior problems for the Helmholtz equation

This paper deals with the coupled procedure of the boundary element method (BEM) and the finite element method (FEM) for the exterior boundary value problems for the Helmholtz equation. A circle is selected as the common boundary on which the integral equation is set up with Fourier expansion. As a result, the exterior problems are transformed into nonlocal boundary value problems in a bounded domain which is treated with FEM, and the normal derivative of the unknown function at the common boundary does not appear. The solvability of the variational equation and the error estimate are also discussed.

     

A finite element procedure of the second order of accuracy

A finite element procedure of the second order of accuracy for solving second order boundary value problems is presented and justified and numerical results are given.