Multiblock mortar expanded mixed formulation for second order nonlinear elliptic problems

Summary form only given. Many materials in the nature or fabricated in laboratories exhibit nonlinear behavior where the permittivity and/or permeability is a function of the magnitude of the electric or magnetic field intensity. Such nonlinearity results in a nonlinear system of Maxwell's equations, which are more challenging to solve. Generally speaking, there are two kinds of methods commonly adopted to solve nonlinear problems. The first kind of methods solves the linearized versions of the original problems, while the second kind of methods deals with the nonlinear problems directly. Although more difficult to solve with a high efficiency, the second kind of methods is more desirable in order to capture the nonlinear phenomenon more accurately. Among various nonlinear solvers developed in the applied mathematics community, the Newton-Raphson method is widely used due to its second-order convergence, if started from an initial guess sufficiently close to the true solution. However, in a real application, there is no straightforward way to choose a good initial guess. As a result, the Newton-Raphson method could be unstable or divergent in solving complicated problems. Another difficulty of applying the Newton-Raphson method is that the Jacobian matrix needs to be updated and solved in every Newton iteration, which could be very time consuming for solving large problems. When the nonlinear electromagnetic problems are formulated and solved with the finite element method in time domain (J. M. Jin, The Finite Element Method in Electromagnetics, Second Edition. New York: Wiley, 2002), the numerical solution at a previous time step can be used as a natural choice for the initial guess of the Newton iteration at the successive time step. The convergent of the Newton-Raphson method can be achieved if the time step is small enough such that the solutions of two successive time steps are sufficiently close. In this presentation, we employ the time-domain finite element method (TDFEM) combined with the Newton-Raphson method and its variations to solve nonlinear electromagnetic problems. We first study the TDFEM formulation of nonlinear problems, and then show how it can be solved with Newton(-like) methods. We demonstrate and compare the performance of different nonlinear solvers including the basic Newton-Raphson method and its variations through a number of numerical examples.

In this paper we study the approximation of solutions to linear and nonlinear elliptic problems with discontinuous coefficients in the discrete duality finite volume (DDFV) framework. This family of schemes allows very general meshes and inherits the main properties of the continuous problem. In order to take into account the discontinuities and to prevent consistency defect in the scheme, we propose to modify the definition of the numerical fluxes on the edges of the mesh where the discontinuity occurs. We first illustrate our approach by the study of the 1D situation. Then, we show how to design our new scheme, called m-DDFV, and we propose its analysis. We also describe an iterative solver, whose convergence is proved, which can be used to solve the nonlinear discrete equations defining the finite volume scheme. Finally, we provide numerical results which confirm that the m-DDFV scheme significantly improves the convergence rate of the usual DDFV method for both linear and nonlinear problems.

We provide a general abstract theory for the solvability and Galerkin approximation of nonlinear twofold saddle point problems. In particular, a Strang error estimate containing the consistency terms arising from the approximation of the continuous operators involved is deduced. Then we apply these results to analyse a fully discrete Galerkin scheme for a twofold saddle point formulation of a nonlinear elliptic boundary value problem in divergence form. Some numerical results are also presented.

BUILDING ELLIPTIC PROBLEM SOLVERS WITH ELLPACK

We first give a stabilized improved moving least squares (IMLS) approximation, which has better computational stability and precision than the IMLS approximation. Then, analysis of the improved element-free Galerkin method is provided theoretically for both linear and nonlinear elliptic boundary value problems. Finally, numerical examples are given to verify the theoretical analysis.

This thesis consists of six research papers.In ``Regularity of the extremal solution in a MEMS model with advection,'' we examine the equation given by $ -\Delta u + c(x) \cdot \nabla u = \lambda f(u) $ in $ \Omega$ with Dirichlet boundary conditions and where $ f(u) = (1-u)^{-2}$ or $ f(u) =e^u$. Our main result is that the associated extremal solution is smooth provided this is the case for the advection free case; $c(x)=0$. In ``Estimates on pull-in distances in MEMS models and other nonlinear eigenvalue problems'' we prove some results, which were observed numerically, regarding equations of the form $ -\Delta u = \lambda |x|^\alpha F(u) $ in $B$ where $B$ is the unit ball in $ \IR^N$. In addition we obtain upper and lower estimates on the extremal solutions associated with various nonlinear eigenvalue problems. In ``The critical dimension for a fourth order elliptic problem with singular nonlinearity,'' we examine the equation given by $ \Delta^2 u = \lambda (1-u)^{-2}$ in $ B$ with Dirichlet boundary conditions where $B$ is the unit ball in $ \IR^N$. Our main result is that the extremal solution $u^*$ is smooth if and only if $ N \le 8$. In ``Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains'' we examine the equation $ \Delta^2 u = \lambda f(u)$ in $ \Omega$ with Navier boundary conditions where $ \Omega$ is a general bounded domain in $ \IR^N$. We obtain various results concerning the regularity of the associated extremal solution. In ``Regularity of the extremal solutions in elliptic systems'' we examine the elliptic system given by $ -\Delta u = \lambda e^v$, \; $ -\Delta v = \gamma e^u$ in $ \Omega$ where $\lambda$ and $\gamma$ are positive constants and we obtain results concering the regularity of the extremal solutions. In ``Optimal Hardy inequalities for general elliptic operators with improvements'' we examine some very general Hardy inequalities. Optimal constants are obtained and we characterize the improvements of these general Hardy inequalities. In addition we prove various weighted versions of these inequalities with improvements and many other results.

There are few numerical techniques available to solve third order boundary value problems. In this paper, we present Galerkin weighted residual method for constructing the numerical solution of third order linear and nonlinear boundary value problems with two point boundary conditions. The method is formulated as a rigorous matrix form. Several numerical examples of both linear and nonlinear boundary value problems in the literature are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods in the literature. Keywords: Galerkin method, Third order linear and nonlinear BVPs, Bernstein Polynomials.

Haar wavelet collocation method for three-dimensional elliptic partial differential equations

As an application of the Schauder Fixed Point Theorem, an existence theory is developed for second order nonlinear differential equations with Stieltjes derivative related to a left-continuous, nondecreasing function. By the method of lower and upper solutions, under a Nagumo-type assumption we get a very general result which can be further applied to deduce the existence of solutions for second order nonlinear problems in the settings of impulsive differential equations, time scale analysis or generalized differential equations.

The Hardy inequality and second order nonlinear eigenvalue problems

In this paper, we find the approximate solution of a second order nonlinear partial differential equation on a simple connected region in R2. We transfer this problem to a new problem of second order non-linear partial differential equation on a rectangle. Then, we transformed the later one to an equivalent optimization problem. Then we consider the optimization problem as a distributed parameter system with artificial controls. Finally, by using the theory of measure, we obtain the approximate solution of the original problem. In this paper also the global error in L1 is controlled.

Introduction.We are here concerned with the oscillatory behavior of solutions of the following second order nonlinear differential equation:where y ^ 1 is the quotient of two odd integers.For equation (2), Jasny [4] and Kurzweil [7] have established the following result on the existence of one oscillatory solution :

On the approximate solutions of a class of fractional order nonlinear Volterra integro-differential initial value problems and boundary value problems of first kind and their convergence analysis

This article continues the derivation of explicit solution and gradient bounds for classes of second order nonlinear parabolic initial-boundary value problems. Earlier results were restricted to domains with boundaries of nonnegative average curvature. The purpose of the present article is to remove the curvature restriction and to extend the results to more general parabolic problems.

Fractional differential equations with strong nonlinearity are important in modelling complex physical phenomena. Therefore, there is an urgent need to employ new technique to help researches and physicists to understand the physical problems better and to deal with such equations. In this paper, we develop and analyze a semi-analytical technique called the homotopy analysis method (HAM) which solve first order fuzzy fractional differential equation with strong nonlinearity under Caputo derivative sense by adjusting the convergence of the series solution. The paper also presents the fuzzy formulation of the proposed HAM using the concepts of fuzzy set theory combined with the properties of fractional calculus and analyzed it in order to obtain the semi-analytical solution. The feasibility of the technique is presented by solving an example of first order nonlinear fractional fuzzy initial value problem (NFFIVP). Finally, the numerical results show that the applied technique is accurate and reliable in solving first order NFFIVP. Thus, the proposed HAM can be extended to treat more general fuzzy fractional order models in physics and engineering.