scholarly journals Analysis of a fully discrete approximation for the classical Keller–Segel model: Lower and a priori bounds

2021 ◽  
Vol 85 ◽  
pp. 69-81
Author(s):  
Juan Vicente Gutiérrez-Santacreu ◽  
José Rafael Rodríguez-Galván
Keyword(s):  
A Priori ◽  
Discrete Approximation ◽  
A Priori Bounds ◽  
Fully Discrete ◽  
Fully Discrete Approximation

scholarly journals Analysis of a Dynamic Viscoelastic Contact Problem with Normal Compliance, Normal Damped Response, and Nonmonotone Slip Rate Dependent Friction

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Mikaël Barboteu ◽  
David Danan
Keyword(s):  
Contact Problem ◽  
Frictional Contact ◽  
Slip Rate ◽  
Approximation Schemes ◽  
Discrete Approximation ◽  
Normal Compliance ◽  
Fully Discrete ◽  
Rate Dependent ◽  
Fully Discrete Approximation ◽  
Normal Damped Response

We consider a mathematical model which describes the dynamic evolution of a viscoelastic body in frictional contact with an obstacle. The contact is modelled with a combination of a normal compliance and a normal damped response law associated with a slip rate-dependent version of Coulomb’s law of dry friction. We derive a variational formulation and an existence and uniqueness result of the weak solution of the problem is presented. Next, we introduce a fully discrete approximation of the variational problem based on a finite element method and on an implicit time integration scheme. We study this fully discrete approximation schemes and bound the errors of the approximate solutions. Under regularity assumptions imposed on the exact solution, optimal order error estimates are derived for the fully discrete solution. Finally, after recalling the solution of the frictional contact problem, some numerical simulations are provided in order to illustrate both the behavior of the solution related to the frictional contact conditions and the theoretical error estimate result.

scholarly journals Quasistatic Porous-Thermoelastic Problems: An a Priori Error Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1436
Author(s):  
Jacobo Baldonedo ◽  
José R. Fernández ◽  
José A. López-Campos
Keyword(s):  
A Priori ◽  
Linear Convergence ◽  
Regularity Conditions ◽  
The Finite Element Method ◽  
Discrete Energy ◽  
Implicit Euler Scheme ◽  
Fully Discrete ◽  
Fully Discrete Approximation ◽  
Thermoelastic Problems ◽  
Discrete Stability

In this paper, we deal with the numerical approximation of some porous-thermoelastic problems. Since the inertial effects are assumed to be negligible, the resulting motion equations are quasistatic. Then, by using the finite element method and the implicit Euler scheme, a fully discrete approximation is introduced. We prove a discrete stability property and a main error estimates result, from which we conclude the linear convergence under appropriate regularity conditions on the continuous solution. Finally, several numerical simulations are shown to demonstrate the accuracy of the approximation, the behavior of the solution and the decay of the discrete energy.

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