Journal ArticleDOI
Monotone explicit iterations of the finite element approximations for the nonlinear boundary value problem
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In this paper, explicit iterations of the finite element schemes for the nonlinear equations associated with the boundary value problem Δu=bu 2, based on piecewise linear polynomials and the lumping operator, are considered.Abstract:
In this paper, we consider monotone explicit iterations of the finite element schemes for the nonlinear equations associated with the boundary value problem Δu=bu 2, based on piecewise linear polynomials and the lumping operator. These iterations construct the monotonically decreasing and increasing sequences, and convergence proofs are given. Finally, we present some numerical examples verifying the effectiveness of the theory.read more
Citations
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Journal ArticleDOI
A Minimax Method for Finding Multiple Critical Points and Its Applications to Semilinear PDEs
Yongxin Li,Jianxin Zhou +1 more
TL;DR: Based on the local theory, a new local numerical minimax method for finding multiple saddle points is developed and implemented successfully to solve a class of semilinear elliptic boundary value problems for multiple solutions on some nonconvex, non star-shaped and multiconnected domains.
Journal ArticleDOI
Algorithms and visualization for solutions of nonlinear elliptic equations
TL;DR: This paper compute and visualize solutions of several major types of semilinear elliptic boundary value problems with a homogeneous Dirichlet boundary condition in 2D using the mountain–pass algorithm (MPA), the scaling iterative algorithm (SIA), the monotone iteration and the direct iteration algorithms (MIA and DIA).
Book ChapterDOI
CHAPTER 3 - Qualitative Properties of Solutions to Elliptic Problems
TL;DR: In this paper, a survey of qualitative properties of elliptic solutions to elliptic equations is presented, focusing on two properties of solutions: the shape of solutions and the stability of solutions.
Journal ArticleDOI
Numerical Methods for Systems of Nonlinear Parabolic Equations with Time Delays
TL;DR: In this paper, the authors investigated the numerical aspects of a class of coupled nonlinear parabolic systems with time delays and showed that the sequence of iterations from each one of these iterative schemes converges monotonically to a unique solution of the finite difference system.
Journal ArticleDOI
Accelerated monotone iterative methods for finite difference equations of reaction-diffusion
TL;DR: In this paper, numerical methods for a finite difference system of reaction-diffusion-convection equation under nonlinear boundary condition are presented, and each of these methods leads to an existence-comparison theorem as well as a computational algorithm for numerical solutions.
References
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Book
Iterative Solution of Nonlinear Equations in Several Variables
J.M. Ortega,Werner C. Rheinboldt +1 more
TL;DR: In this article, the authors present a list of basic reference books for convergence of Minimization Methods in linear algebra and linear algebra with a focus on convergence under partial ordering.
Book
Matrix iterative analysis
TL;DR: In this article, the authors propose Matrix Methods for Parabolic Partial Differential Equations (PPDE) and estimate of Acceleration Parameters, and derive the solution of Elliptic Difference Equations.
Journal ArticleDOI
Maximum principle and uniform convergence for the finite element method
TL;DR: In this paper, it was shown that the Dirichlet boundary value problem converges uniformly to the exact solution u if u ϵ W1,p (Ω), with p > n, and that ∥u−u h ∥ L ∞(Ω) = O(h) if uϵ W2,p(ϵ), with 2p > n.