Analysis of multiscale mortar mixed approximation of nonlinear elliptic equations

A multiscale mortar mixed finite element method is established to approximate non-linear second order elliptic equations. The method is based on non-overlapping domain decomposition and mortar finite element methods. The existence and uniqueness of the approximation are demonstrated, and a priori $L^2$-error estimates for the velocity and pressure are derived. An error bound for mortar pressure is proved. Convergence estimates of the mortar pressure are based on a linear interface formulation having the discrete-pressure dependent coefficient. Optimal order convergence is achieved on the fine scale by a proper choice of mortar space and polynomial degree of approximation. The quadratic convergence of the Newton–Raphson method is proved for the nonlinear algebraic system arising from the mortar mixed formulation of the problem. Numerical experiments are performed to support theoretic results.     

Mathematical model and computer simulation of three dimensional thin film elliptic interface problems

Many engineering applications require information about temperature distribution in multilayer thin films. Steady state heat conduction in multilayer structures can be modelled by elliptic interface problems. In this paper, computer simulations of some thin film elliptic interface problems have been performed by applying decomposed immersed interface method on MATLAB.     

Legendre–Galerkin spectral approximation for a nonlocal elliptic Kirchhof-type problem

We propose a numerical scheme to obtain an approximate solution of a nonlocal elliptic Kirchhof-type problem. We first reduce the problem to a nonlinear finite dimensional system by a Legendre–Galerkin spectral method and then solve it by an iterative process. Convergence of the iterative process and an error estimation of the approximate solution is provided. Numerical experiments are conducted to illustrate the performance of the proposed method.     

A multigrid scheme for 3D Monge–Ampère equations*

The elliptic Monge–Ampère equation is a fully nonlinear partial differential equation which has been the focus of increasing attention from the scientific computing community. Fast three-dimensional solvers are needed, for example in medical image registration but are not yet available. We build fast solvers for smooth solutions in three dimensions using a nonlinear full-approximation storage multigrid method. Starting from a second-order accurate centred finite difference approximation, we present a nonlinear Gauss–Seidel iterative method which has a mechanism for selecting the convex solution of the equation. The iterative method is used as an effective smoother, combined with the full-approximation storage multigrid method. Numerical experiments are provided to validate the accuracy of the finite difference scheme and illustrate the computational efficiency of the proposed multigrid solver.     

A high-order compact scheme for the one-dimensional Helmholtz equation with a discontinuous coefficient

In this paper, based on the idea of the immersed interface method, a fourth-order compact finite difference scheme is proposed for solving one-dimensional Helmholtz equation with discontinuous coefficient, jump conditions are given at the interface. The Dirichlet boundary condition and the Neumann boundary condition are considered. The Neumann boundary condition is treated with a fourth-order scheme. Numerical experiments are included to confirm the accuracy and efficiency of the proposed method.     

Simultaneous identification of Robin coefficient and heat flux in an elliptic system

In this paper, we shall derive and propose an efficient algorithm for simultaneously reconstructing the Robin coefficient and heat flux in an elliptic system from part of the boundary measurements. The uniqueness of the simultaneous identification is demonstrated. The ill-posed inverse problem is formulated into an output least-squares nonlinear and non-convex minimization with Tikhonov regularization, while the regularizing effects of the regularized system are justified. The Levenberg–Marquardt method is applied to change the non-convex minimization into convex minimization, which will be solved by surrogate functional method so as to get the explicit expression of the minimizer. Numerical experiments are provided to show the accuracy and efficiency of the algorithm.     

Monotone Iterates for Solving Nonlinear Monotone Difference Schemes

This paper is concerned with monotone iterative algorithms for solving nonlinear monotone difference schemes of elliptic type. Firstly, the monotone method (known as the method of lower and upper solutions) is applied to computing the nonlinear monotone difference schemes in the canonical form. Secondly, a monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. This monotone algorithm solves only linear discrete systems at each iterative step and converges monotonically to the exact solution of the nonlinear monotone difference schemes. Numerical experiments are presented.     

Numerical Solution of a Reaction-Diffusion Elliptic Interface Problem with Strong Anisotropy

We consider a singularly perturbed reaction-diffusion elliptic problem in two dimensions (x,y), with strongly anisotropic coefficients and line interface. The second order derivative with respect to x is multiplied by a small parameter ². We construct finite volume difference schemes on condensed Shihskin meshes and prove -uniform convergence in discrete energy and maximum norms. Numerical experiments that agree with the theoretical results are given.     

求解带有间断系数泊松方程的修正有限体积

利用修正的有限体积方法求解带有间断系数的泊松方程,改进是对基于笛卡尔坐标系下的调和平均系数进行的。数值实验表明新格式二阶逐点收敛并且在界面处具有二阶精度,新方法较已有的求解不连续扩散系数的算术平均法和调和平均法,特别是在系数跳跃较大的情况下更具优势。     

Fitted strong stability-preserving schemes for the Black-Scholes-Barenblatt equation

We solve numerically a fully nonlinear Black–Scholes problem of Bellman type. The algorithm is focused on the so-called Delta greek, the first spatial derivative of the option price. Since the elliptic operator degenerates on the boundary we use a fitted finite volume discretization in space. Strong stability-preserving time-marching is further applied in accordance to the nonlinear nature of the differential problem. Numerical experiments validate our considerations.     

Mixed discontinuous Legendre wavelet Galerkin method for solving elliptic partial differential equations

By incorporating the Legendre multiwavelet into the mixed discontinuous Galerkin method, in this paper, we present a novel method for solving second-order elliptic partial differential equations (PDEs), which is known as the mixed discontinuous Legendre multiwavelet Galerkin method, derive an adaptive algorithm for the method and estimate the approximating error of its numerical fluxes. One striking advantage of our method is that the differential operator, boundary conditions and numerical fluxes involved in the elementwise computation can be done with lower time cost. Numerical experiments demonstrate the validity of this method. The proposed method is also applicable to some other kinds of PDEs.     

A semi-explicit multi-symplectic splitting scheme for a 3-coupled nonlinear Schrödinger equation

In this paper, we mainly propose an efficient semi-explicit multi-symplectic splitting scheme to solve a 3-coupled nonlinear Schrödinger (3-CNLS) equation. Based on its multi-symplectic formulation, the 3-CNLS equation can be split into one linear multi-symplectic subsystem and one nonlinear infinite-dimensional Hamiltonian subsystem. For the linear subsystem, the multi-symplectic Fourier pseudospectral method and symplectic Euler method are employed in spatial and temporal discretizations, respectively. For the nonlinear subsystem, the mid-point symplectic scheme is used. Numerical experiments for the unstable plane waves show the effectiveness of the proposed method during long-time numerical calculation.     

A Finite Difference Method and Analysis for 2D Nonlinear Poisson–Boltzmann Equations

A fast finite difference method based on the monotone iterative method and the fast Poisson solver on irregular domains for a 2D nonlinear Poisson–Boltzmann equation is proposed and analyzed in this paper. Each iteration of the monotone method involves the solution of a linear equation in an exterior domain with an arbitrary interior boundary. A fast immersed interface method for generalized Helmholtz equations on exterior irregular domains is used to solve the linear equation. The monotone iterative method leads to a sequence which converges monotonically from either above or below to a unique solution of the problem. This monotone convergence guarantees the existence and uniqueness of a solution as well as the convergence of the finite difference solution to the continuous solution. A comparison of the numerical results against the exact solution in an example indicates that our method is second order accurate. We also compare our results with available data in the literature to validate the numerical method. Our method is efficient in terms of accuracy, speed, and flexibility in dealing with the geometry of the domain     

Optimized Schwarz methods without overlap for highly heterogeneous media

In this paper an original variant of the Schwarz domain decomposition method is introduced for heterogeneous media. This method uses new optimized interface conditions specially designed to take into account the heterogeneity between the sub-domains on each sides of the interfaces. Numerical experiments illustrate the dependency of the proposed method with respect to several parameters, and confirm the robustness and efficiency of this method based on such optimized interface conditions. Several mesh partitions taking into account multiple cross points are considered in these experiments.     

基于混合遗传算法求解非线性方程组

将非线性方程组的求解问题转化为函数优化问题,且综合考虑了拟牛顿法和遗传算法各自的优点,提出了一种用于求解非线性方程组的混合遗传算法。该混合算法充分发挥了拟牛顿法的局部搜索、收敛速度快和遗传算法的群体搜索、全局收敛的优点。为了证明该混合遗传算法的有效性,选择了几个典型的非线性方程组,从实验计算结果、收敛可靠性指标对比不同算法进行分析。数值模拟实验表明,该混合遗传算法具有很高的精确性和收敛性,是求解非线性方程组的一种有效算法。     

A simple weak formulation for solving two-dimensional diffusion equation with local reaction on the interface

In this paper, we propose a numerical method for solving two-dimensional diffusion equation with nonhomogeneous jump condition and nonlinear flux jump condition located at the interface. We use finite element method coupled with Newton’s method to deal with the jump conditions and to linearize the system. It is easy to implement. The grid used here is body-fitting grids based on the idea of semi-Cartesian grid. Numerical experiments show that this method is nearly second order accurate in the $L^{\infty }$ norm.     

A New Approach for Numerically Solving Nonlinear Eigensolution Problems

By considering a constraint on the energy profile, a new implicit approach is developed to solve nonlinear eigensolution problems. A corresponding minimax method is modified to numerically find eigensolutions in the order of their eigenvalues to a class of semilinear elliptic eigensolution problems from nonlinear optics and other nonlinear dispersive/diffusion systems. It turns out that the constraint is equivalent to a constraint on the wave intensity in L-(p+1) norm. The new approach enables people to establish some interesting new properties, such as wave intensity preserving/control, bifurcation identification, etc., and to explore their applications. Numerical results are presented to illustrate the method.     

An implicit enthalpy scheme for one-phase Stefan problems

A compact finite-difference scheme to solve one-phase Stefan problems in one dimension is described. Numerical experiments indicate that the moving interface is obtained withO(t) accuracy when 3–4 iterations per time step are used to solve the nonlinear implicit scheme. The scheme can be adapted to ADI methods in higher dimensions.     

A Non-Overlapping Domain Decomposition Method for the Advection-Diffusion Problem

The application of a non-overlapping domain decomposition method to the solution of a stabilized finite element method for elliptic boundary value problems is considered. We derive an a-posteriori error estimate which bounds the error on the subdomains by the interface error of the subdomain solutions. As a by-product, some foundation is given to the design of the interface transmission condition. Numerical results support the theoretical results. Furthermore, we adapt a recent result on a-posteriori estimates for singular perturbation problems in order to obtain an a-posteriori estimate for the discrete subdomain solutions.     

双曲方程基于分离算子的交替方向差分算法

因为在自然科学领域有着广泛的应用,双曲型方程组的数值求解一直是研究的热点．本文中,为求解一类非线性二阶双曲型方程,将方程中的非线性椭圆微分算子分解为线性部分和非线性部分,对线性部分用隐格式逼近,对非线性部分用显格式逼近,这种方法可以把非线性问题转化成每一时间层只有右端项不同的线性方程组,计算简单且计算格式绝对稳定;交替方向格式可以把多维问题转化成一维问题,x,y两个方向的迭代矩阵均为三对角矩阵,结构相同,易于编程并行计算．最后通过数值实验表明结果符合理论分析．