Numerical solutions of coupled systems of nonlinear elliptic equations

This article deals with numerical solutions of a general class of coupled nonlinear elliptic equations. Using the method of upper and lower solutions, monotone sequences are constructed for difference schemes which approximate coupled systems of nonlinear elliptic equations. This monotone convergence leads to existence‐uniqueness theorems for solutions to problems with reaction functions of quasi‐monotone nondecreasing, quasi‐monotone nonincreasing and mixed quasi‐monotone types. A monotone domain decomposition algorithm which combines the monotone approach and an iterative domain decomposition method based on the Schwarz alternating, is proposed. An application to a reaction‐diffusion model in chemical engineering is given. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 621–640, 2012     

Uniform quadratic convergence of monotone iterates for nonlinear singularly perturbed parabolic problems

This paper deals with a monotone iterative method for solving nonlinear singularly perturbed parabolic problems. Monotone sequences, based on the method of upper and lower solutions, are constructed for a nonlinear difference scheme which approximates the nonlinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. The monotone sequences possess quadratic convergence rate. An analysis of uniform convergence of the monotone iterative method to the solutions of the nonlinear difference scheme and to the continuous problem is given. Numerical experiments are presented.     

求解一类反应扩散方程组数值解的组合单调迭代法

给出一类求解带非线性边界条件的反应扩散方程组的组合单调迭代法,证明了当反应项和边界条件具有拟单调性和迭代充阢的单调收敛性以及数值方法的稳定性。     

A modified accelerated monotone iterative method for finite difference reaction-diffusion-convection equations

This paper is concerned with monotone algorithms for the finite difference solutions of a class of nonlinear reaction-diffusion-convection equations with nonlinear boundary conditions. A modified accelerated monotone iterative method is presented to solve the finite difference systems for both the time-dependent problem and its corresponding steady-state problem. This method leads to a simple and yet efficient linear iterative algorithm. It yields two sequences of iterations that converge monotonically from above and below, respectively, to a unique solution of the system. The monotone property of the iterations gives concurrently improving upper and lower bounds for the solution. It is shown that the rate of convergence for the sum of the two sequences is quadratic. Under an additional requirement, quadratic convergence is attained for one of these two sequences. In contrast with the existing accelerated monotone iterative methods, our new method avoids computing local maxima in the construction of these sequences. An application using a model problem gives numerical results that illustrate the effectiveness of the proposed method.     

MONOTONE APPROXIMATION TO A SYSTEM WITHOUT MONOTONE NONLINEARITY

1. IntroductionDue to the development of various studies in electromagnetism, biology and someother fields, nonlinear systems hajve been paid extellsive attention both analytically andnumerically, e.g., see [1--12]. As we kll')w, a reasonable numerical method should notonly have the approximation error of higher order, but also preserve the main feature ofthe original problem. In this case? the numerical results might fit the physical processbetter. Since the usual approximations simulate the …     

Uniform convergent monotone iterates for semilinear singularly perturbed parabolic problems

This paper deals with discrete monotone iterative methods for solving semilinear singularly perturbed parabolic problems. Monotone sequences, based on the accelerated monotone iterative method, are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. An analysis of uniform convergence of the monotone iterative method to the solutions of the nonlinear difference scheme and continuous problem is given. Numerical experiments are presented.     

Monotone iterates with quadratic convergence rate for solving weighted average approximations to semilinear parabolic problems

This paper deals with discrete monotone iterative methods for solving semilinear singularly perturbed parabolic problems. Monotone sequences, based on the accelerated monotone iterative method, are constructed for a nonlinear difference scheme which approximates the semilinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. An analysis of convergence of the monotone iterative method to the solutions of the nonlinear difference scheme is given. Numerical experiments are presented.     

Yosida approximations for multivalued stochastic partial differential equations driven by Lévy noise on a Gelfand triple

We prove the existence and uniqueness of solutions for a class of multivalued stochastic partial differential equations with maximal monotone drift on Banach space driven by multiplicative Lévy noise. We also establish the strong convergence result for solutions of the approximating equations where the maximal monotone drift operator is replaced by its Yosida approximation. As an application, the existence and uniqueness of solutions for multivalued stochastic porous medium equations is obtained.     

Tight global linear convergence rate bounds for Douglas–Rachford splitting

Recently, several authors have shown local and global convergence rate results for Douglas–Rachford splitting under strong monotonicity, Lipschitz continuity, and cocoercivity assumptions. Most of these focus on the convex optimization setting. In the more general monotone inclusion setting, Lions and Mercier showed a linear convergence rate bound under the assumption that one of the two operators is strongly monotone and Lipschitz continuous. We show that this bound is not tight, meaning that no problem from the considered class converges exactly with that rate. In this paper, we present tight global linear convergence rate bounds for that class of problems. We also provide tight linear convergence rate bounds under the assumptions that one of the operators is strongly monotone and cocoercive, and that one of the operators is strongly monotone and the other is cocoercive. All our linear convergence results are obtained by proving the stronger property that the Douglas–Rachford operator is contractive.     

Parameter identification in nonlocal nonlinear evolution equations

We develop an approximation framework for identifying parameters in a general class of nonautonomous, nonlocal and nonlinear evolution equations. After establishing existence and uniqueness of solutions, we present a convergence theory for Galerkin approximations to inverse problems involving these equations. Our approach relies on the theory of maximal monotone operators in Banach spaces. An application to a nonautonomous nonlinear integral equation arising in heat flow is also discussed.     

Existence of Solutions to a Class of Fractional Differential Equations

In this paper, the existence of solutions to a class of fractional differential equations $D_{0+}^{\alpha}u(t)=h(t)f(t, u(t), D_{0+}^{\theta}u(t))$ is obtained by an efficient and simple monotone iteration method. At first, the existence of a solution to the problem above is guaranteed by finding a bounded domain $D_M$ on functions $f$ and $g$. Then, sufficient conditions for the existence of monotone solution to the problem are established by applying monotone iteration method. Moreover, two efficient iterative schemes are proposed, and the convergence of the iterative process is proved by using the monotonicity assumption on $f$ and $g$. In particular, a new algorithm which combines Gauss-Kronrod quadrature method with cubic spline interpolation method is adopted to achieve the monotone iteration method in Matlab environment, and the high-precision approximate solution is obtained. Finally, the main results of the paper are illustrated by some numerical simulations, and the approximate solutions graphs are provided by using the iterative method.     

On approximations of solutions of stochastic equations with monotone coefficients

An estimate is obtained for the rate of convergence for step function approximations of solutions of stochastic differential equations with monotone coefficients.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 25–28, 1986.     

Computational error analysis of periodic solutions for singular semilinear parabolic problems

By using the monotone method, a theoretical and computational method is given to find, to the degree of accuracy desired, approximate solutions of a class of singular semilinear parabolic problems. So that the error between the actual solution and its approximation is within a given error tolerance, the number of iterations is determined. Since each iterate is in terms of an infinite series, the number of terms to be retained in each iterate is determined so that its error from the exact iterate is within a given error tolerance. An improved rate of convergence is then given to show that it is possible to reduce the number of terms retained in each iterate. An algorithm is also described to obtain numerical solutions. For illustration of the computational methods developed, a numerical example is given.     

一类非线性边界条件的抛物型方程组的周期解的数值解法

本文研究了一类具有非线性边界条件的反应一扩散一对流方程组的周期解的数值解法，利用上下解作为初始迭代，把求方程组的Jacobi方法和Gauss—Seidel方法和上下解方法结合起来，得到了迭代序列的单调收敛性和方法的收敛性，对方法的稳定性也作了论述。     

Monotone convergence of finite element approximations of obstacle problems

The purpose of the work is to study the monotone convergence of numerical solutions of obstacle problems under mesh refinement when the obstacle is convex. We prove monotone convergence of piecewise linear finite element approximations for one-dimensional obstacle problems. We demonstrate by giving a example that such monotone convergence will not hold in the two-dimensional case.     

On a Gradient Flow with Exponential Rate of Convergence

We present an evolution equation governed by a maximal monotone operator with exponential rate of convergence to a zero of the maximal monotone operator. When the maximal monotone operator is the subdifferential of a convex, proper, and lower semicontinuous function, we show that the trajectory of solutions of the evolution equation converges exponentially to the minimum value of the convex function.     

Monotone convergence theorems for Henstock-Kurzweil integrable functions and applications

In this paper we prove monotone convergence theorems for Henstock-Kurzweil integrable functions from a compact real interval to an ordered Banach space. These theorems are then applied to prove existence results for solutions of a discontinuous functional integral equation containing Henstock-Kurzweil integrable functions.     

Strongly coupled elliptic systems and applications to Lotka–Volterra models with cross-diffusion

The aim of this paper is to investigate the existence and method of construction of solutions for a general class of strongly coupled elliptic systems by the method of upper and lower solutions and its associated monotone iterations. The existence problem is for nonquasimonotone functions arising in the system, while the monotone iterations require some mixed monotone property of these functions. Applications are given to three Lotka–Volterra model problems with cross-diffusion and self-diffusion which are some extensions of the classical competition, prey–predator, and cooperating ecological systems. The monotone iterative schemes lead to some true positive solutions of the competition system, and to quasisolutions of the prey–predator and cooperating systems. Also given are some sufficient conditions for the existence of a unique positive solution to each of the three model problems.     

Entropy solution theory for fractional degenerate convection–diffusion equations

We study a class of degenerate convection-diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection-diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Lévy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.     

Iterative methods for solving monotone equilibrium problems via dual gap functions

This paper proposes an iterative method for solving strongly monotone equilibrium problems by using gap functions combined with double projection-type mappings. Global convergence of the proposed algorithm is proved and its complexity is estimated. This algorithm is then coupled with the proximal point method to generate a new algorithm for solving monotone equilibrium problems. A class of linear equilibrium problems is investigated and numerical examples are implemented to verify our algorithms.