scholarly journals Wavelet Numerical Solutions for a Class of Elliptic Equations with Homogeneous Boundary Conditions

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1381
Author(s):  
Jinru Wang ◽  
Wenhui Shi ◽  
Lin Hu
Keyword(s):  
Elliptic Equations ◽  
Numerical Solutions ◽  
Homogeneous Boundary Condition ◽  
Riesz Bases ◽  
Condition Numbers ◽  
Order Elliptic Equation ◽  
Approximation Properties ◽  
Numerical Examples ◽  
Stiffness Matrices ◽  
Second Order Elliptic Equation

This paper focuses on a method to construct wavelet Riesz bases with homogeneous boundary condition and use them to a kind of second-order elliptic equation. First, we construct the splines on the interval [0,1] and consider their approximation properties. Then we define the wavelet bases and illustrate the condition numbers of stiffness matrices are small and bounded. Finally, several numerical examples show that our approach performs efficiently.

scholarly journals ASYMPTOTIC BEHAVIOR OF BLOWUP SOLUTIONS FOR ELLIPTIC EQUATIONS WITH EXPONENTIAL NONLINEARITY AND SINGULAR DATA

2009 ◽  
Vol 11 (03) ◽  
pp. 395-411 ◽  
Author(s):  
LEI ZHANG
Keyword(s):  
Elliptic Equations ◽  
Type Equation ◽  
Global Solutions ◽  
Order Elliptic Equation ◽  
Liouville Type ◽  
Exponential Nonlinearity ◽  
Second Order Elliptic Equation ◽  
Singular Data ◽  
Sharp Error ◽  
Uniformly Bounded

We consider a sequence of blowup solutions of a two-dimensional, second-order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of Bartolucci–Chen–Lin–Tarantello, it is proved that the profile of the solutions differs from global solutions of a Liouville-type equation only by a uniformly bounded term. The present paper improves their result and establishes an expansion of the solutions near the blowup points with a sharp error estimate.

MORTAR SPECTRAL ELEMENT METHODS FOR ELLIPTIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

2002 ◽  
Vol 12 (04) ◽  
pp. 497-524 ◽  
Author(s):  
CHRISTINE BERNARDI ◽  
NEJMEDDINE CHORFI
Keyword(s):  
Elliptic Equations ◽  
Spectral Element ◽  
Discontinuous Coefficients ◽  
Optimal Error Estimates ◽  
Order Elliptic Equation ◽  
Optimal Error ◽  
Element Discretization ◽  
Second Order Elliptic Equation ◽  
Piecewise Continuous ◽  
Dimensional Domain

We consider a second-order elliptic equation with piecewise continuous coefficients in a bounded two-dimensional domain. We propose a spectral element discretization of this problem which relies on the mortar domain decomposition technique. We prove optimal error estimates. Next, we compare several versions, conforming or not, of this discretization by means of numerical experiments.

Global regularity estimates for non-divergence elliptic equations on weighted variable Lebesgue spaces

2020 ◽  
pp. 2050014
Author(s):  
The Quan Bui ◽  
The Anh Bui ◽  
Xuan Thinh Duong
Keyword(s):  
Elliptic Equations ◽  
Global Regularity ◽  
Order Elliptic Equation ◽  
Lebesgue Spaces ◽  
Variable Lebesgue Spaces ◽  
Regularity Estimates ◽  
Sparse Operators ◽  
Second Order Elliptic Equation ◽  
Bmo Coefficients ◽  
Weighted Variable

This paper is to prove global regularity estimates for solutions to the second-order elliptic equation in non-divergence form with BMO coefficients in a [Formula: see text] domain on weighted variable exponent Lebesgue spaces. Our approach is based on the representations for the solutions to the non-divergence elliptic equations and the domination technique by sparse operators in harmonic analysis.

scholarly journals Estimates for solutions of elliptic equations in a limit case

1987 ◽  
Vol 36 (3) ◽  
pp. 425-434 ◽  
Author(s):  
Ester Giarrusso ◽  
Guido Trombetti
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equations ◽  
Divergence Form ◽  
Limit Case ◽  
Order Elliptic Equation ◽  
Best Constant ◽  
Homogeneous Dirichlet Problem ◽  
Second Order Elliptic Equation ◽  
Bounded Open Subset ◽  
The Right

Let u be a week solution of homogeneous Dirichlet problem for a second order elliptic equation of divergence form, in a bounded open subset of ℝn. We prove, that if the right hand side of the equation is an element of H−1, n(Ω), then u belongs to the Orlicz space LΦ where Φ(t) = exp(|t|n/(n−1)) − 1. We employ the properties of the Schwartz symmetrization thus obtaining the “best” constant of the estimate.

A stochastic operational matrix method for numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations

2020 ◽  
Vol 28 (3) ◽  
pp. 209-216
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Numerical Examples ◽  
Operational Matrix Method ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Block Pulse Function

AbstractIn this article, hybrid Legendre block-pulse functions are implemented in determining the approximate solutions for multi-dimensional stochastic Itô–Volterra integral equations. The block-pulse function and the proposed scheme are used for deriving a methodology to obtain the stochastic operational matrix. Error and convergence analysis of the scheme is discussed. A brief discussion including numerical examples has been provided to justify the efficiency of the mentioned method.

Perturbation Bounds and Condition Numbers for a Complex Indefinite Linear Algebraic System

2016 ◽  
Vol 6 (2) ◽  
pp. 211-221 ◽  
Author(s):  
Lei Zhu ◽  
Wei-Wei Xu ◽  
Xing-Dong Yang
Keyword(s):  
Algebraic System ◽  
Condition Numbers ◽  
Linear Algebraic System ◽  
Science And Engineering ◽  
Numerical Examples ◽  
Interest In Science ◽  
Perturbation Bounds

AbstractWe consider perturbation bounds and condition numbers for a complex indefinite linear algebraic system, which is of interest in science and engineering. Some existing results are improved, and illustrative numerical examples are provided.

scholarly journals Local Elastic and Geometric Stiffness Matrices for the Shell Element Applied in cFEM

2017 ◽  
Author(s):  
Dávid Visy ◽  
Sándor Ádány
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Shell Element ◽  
Numerical Examples ◽  
Geometric Stiffness ◽  
Stiffness Matrices ◽  
Plane Element ◽  
Unusual Combination ◽  
Shell Finite Element ◽  
Lagrange Strain

In this paper local elastic and geometric stiffness matrices of ashell finite element are presented and discussed. The shell finiteelement is a rectangular plane element, specifically designedfor the so-called constrained finite element method. One of themost notable features of the proposed shell finite element isthat two perpendicular (in-plane) directions are distinguished,which is resulted in an unusual combination of otherwise classicshape functions. An important speciality of the derived stiffnessmatrices is that various options are considered, whichallows the user to decide how to consider the through-thicknessstress-strain distributions, as well as which second-order strainterms to consider from the Green-Lagrange strain matrix. Thederivations of the stiffness matrices are briefly summarizedthen numerical examples are provided. The numerical examplesillustrate the effect of the various options, as well as theyare used to prove the correctness of the proposed shell elementand of the completed derivations.

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