scholarly journals General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1599
Author(s):  
Stoil I. Ivanov
Keyword(s):  
Error Estimates ◽  
Local Convergence ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Sufficient Conditions ◽  
Posteriori Error ◽  
Picard Iteration ◽  
Polynomial Zeros ◽  
Convergence Theorems ◽  
Simple And Multiple Roots

In this paper, we prove two general convergence theorems with error estimates that give sufficient conditions to guarantee the local convergence of the Picard iteration in arbitrary normed fields. Thus, we provide a unified approach for investigating the local convergence of Picard-type iterative methods for simple and multiple roots of nonlinear equations. As an application, we prove two new convergence theorems with a priori and a posteriori error estimates about the Super-Halley method for multiple polynomial zeros.

scholarly journals Error estimates for approximating best proximity points for cyclic contractive maps

2016 ◽  
Vol 32 (2) ◽  
pp. 265-270 ◽  
Author(s):  
BOYAN ZLATANOV ◽  
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Convex Banach Space ◽  
Picard Iteration ◽  
Power Type ◽  
Cyclic Contraction ◽  
Contraction Map ◽  
Contractive Maps

We find a priori and a posteriori error estimates of the best proximity point for the Picard iteration associated to a cyclic contraction map, which is defined on a uniformly convex Banach space with modulus of convexity of power type.

scholarly journals A Priori and A Posteriori Error Estimates for a Crank Nicolson Type Scheme of an Elliptic Problem with Dynamical Boundary Conditions

2016 ◽  
Vol 8 (2) ◽  
pp. 1
Author(s):  
Rola Ali Ahmad ◽  
Toufic El Arwadi ◽  
Houssam Chrayteh ◽  
Jean-Marc Sac-Epee
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Time Discretization ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Dynamical Boundary Conditions ◽  
Error Indicators ◽  
Crank Nicolson

In this article we claim that we are going to give a priori and a posteriori error estimates for a Crank Nicolson type scheme. The problem is discretized by the finite elements in space. The main result of this paper consists in establishing two types of error indicators, the first one linked to the time discretization and the second one to the space discretization.

ERROR ESTIMATES FOR LINEAR PDEs SOLVED BY WAVELET BASED TAYLOR–GALERKIN SCHEMES

2009 ◽  
Vol 07 (02) ◽  
pp. 143-162 ◽  
Author(s):  
MANI MEHRA ◽  
B. V. RATHISH KUMAR
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Equivalent Norm ◽  
Posteriori Error ◽  
Discrete Problem ◽  
Fully Discrete ◽  
Time Stepping ◽  
Priori Estimates ◽  
Space Error

In this paper, we develop a priori and a posteriori error estimates for wavelet-Taylor–Galerkin schemes introduced in Refs. 6 and 7 (particularly wavelet Taylor–Galerkin scheme based on Crank–Nicolson time stepping). We proceed in two steps. In the first step, we construct the priori estimates for the fully discrete problem. In the second step, we construct error indicators for posteriori estimates with respect to both time and space approximations in order to use adaptive time steps and wavelet adaptivity. The space error indicator is computed using the equivalent norm expressed in terms of wavelet coefficients.

scholarly journals On the Best Proximity Points for p–Cyclic Summing Contractions

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1060
Author(s):  
Miroslav Hristov ◽  
Atanas Ilchev ◽  
Boyan Zlatanov
Keyword(s):  
Fixed Point Theory ◽  
Complete Metric Space ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Point Theory ◽  
Picard Iteration ◽  
Best Proximity Points ◽  
Underlying Space ◽  
Cyclic Maps

We present a condition that guarantees the existence and uniqueness of fixed (or best proximity) points in complete metric space (or uniformly convex Banach spaces) for a wide class of cyclic maps, called p–cyclic summing maps. These results generalize some known results from fixed point theory. We find a priori and a posteriori error estimates of the fixed (or best proximity) point for the Picard iteration associated with the investigated class of maps, provided that the modulus of convexity of the underlying space is of power type. We illustrate the results with some applications and examples.

scholarly journals CONVERGENCE AND ERROR ESTIMATES OF TWO ITERATIVE METHODS FOR THE STRONG SOLUTION OF THE INCOMPRESSIBLE KORTEWEG MODEL

2009 ◽  
Vol 19 (09) ◽  
pp. 1713-1742
Author(s):  
F. GUILLÉN-GONZÁLEZ ◽  
M. A. RODRÍGUEZ-BELLIDO
Keyword(s):  
Strong Solution ◽  
Error Estimates ◽  
Fluid Model ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Strong Solutions ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Banach’S Fixed Point Theorem

We show the existence of strong solutions for a fluid model with Korteweg tensor, which is obtained as limit of two iterative linear schemes. The different unknowns are sequentially decoupled in the first scheme and in parallel form in the second one. In both cases, the whole sequences are bounded in strong norms and convergent towards the strong solution of the system, by using a generalization of Banach's fixed point theorem. Moreover, we explicit a priori and a posteriori error estimates (respect to the weak norms), which let us to compare both schemes.

DISCRETIZATION OF AN UNSTEADY FLOW THROUGH A POROUS SOLID MODELED BY DARCY'S EQUATIONS

2008 ◽  
Vol 18 (12) ◽  
pp. 2087-2123 ◽  
Author(s):  
CHRISTINE BERNARDI ◽  
VIVETTE GIRAULT ◽  
KUMBAKONAM R. RAJAGOPAL
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Convergence Properties ◽  
A Posteriori Error ◽  
Flow Through ◽  
Priori Error Estimates ◽  
Dependent Flow ◽  
Darcy’S Equations

The system of unsteady Darcy's equations considered here models the time-dependent flow of an incompressible fluid such as water in a rigid porous medium. We propose a discretization of this problem that relies on a backward Euler's scheme for the time variable and finite elements for the space variables. We prove a priori error estimates that justify the optimal convergence properties of the discretization and a posteriori error estimates that allow for an efficient adaptivity strategy both for the time steps and the meshes.

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