scholarly journals Analysis of fully discrete, quasi non-conforming approximations of evolution equations and applications

2021 ◽  
pp. 1-47
Author(s):  
Luigi C. Berselli ◽  
Alex Kaltenbach ◽  
Michael Růžička
Keyword(s):  
Final Section ◽  
Evolution Equations ◽  
Monotone Operators ◽  
Incompressible Fluids ◽  
Discrete Approximations ◽  
Evolution Problems ◽  
Fully Discrete ◽  
Abstract Evolution Equations ◽  
Spatial Approximation ◽  
Continuous Problem

In this paper, we consider fully discrete approximations of abstract evolution equations, by means of a quasi non-conforming spatial approximation and finite differences in time (Rothe–Galerkin method). The main result is the convergence of the discrete solutions to a weak solution of the continuous problem. Hence, the result can be interpreted either as a justification of the numerical method or as an alternative way of constructing weak solutions. We set the problem in the very general and abstract setting of pseudo-monotone operators, which allows for a unified treatment of several evolution problems. The examples — which fit into our setting and which motivated our research — are problems describing the motion of incompressible fluids, since the quasi non-conforming approximation allows to handle problems with prescribed divergence. Our abstract results for pseudo-monotone operators allow to show convergence just by verifying a few natural assumptions on the operator time-by-time and on the discretization spaces. Hence, applications and extensions to several other evolution problems can be easily performed. The results of some numerical experiments are reported in the final section.

scholarly journals On the frictionless unilateral contact of two viscoelastic bodies

2003 ◽  
Vol 2003 (11) ◽  
pp. 575-603 ◽  
Author(s):  
M. Barboteu ◽  
T.-V. Hoarau-Mantel ◽  
M. Sofonea
Keyword(s):  
Evolution Equations ◽  
Maximal Monotone ◽  
Viscoelastic Behavior ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Constitutive Law ◽  
Test Problems ◽  
Fully Discrete ◽  
Deformable Bodies ◽  
Augmented Lagrangian Approach

We consider a mathematical model which describes the quasistatic contact between two deformable bodies. The bodies are assumed to have a viscoelastic behavior that we model with Kelvin-Voigt constitutive law. The contact is frictionless and is modeled with the classical Signorini condition with zero-gap function. We derive a variational formulation of the problem and prove the existence of a unique weak solution to the model by using arguments of evolution equations with maximal monotone operators. We also prove that the solution converges to the solution of the corresponding elastic problem, as the viscosity tensors converge to zero. We then consider a fully discrete approximation of the problem, based on the augmented Lagrangian approach, and present numerical results of two-dimensional test problems.

Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

2021 ◽  
Vol 19 (1) ◽  
pp. 111-120
Author(s):  
Qinghua Zhang ◽  
Zhizhong Tan
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Nonlinear Operator ◽  
Evolution Equations ◽  
Bounded Function ◽  
Inhomogeneous Term ◽  
Linear Evolution ◽  
Abstract Evolution Equations ◽  
Linear Evolution Equation

Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

Full discretisation of second-order nonlinear evolution equations: strong convergence and applications

2011 ◽  
Vol 11 (4) ◽  
pp. 441-459 ◽  
Author(s):  
Etienne Emmrich ◽  
David Šiška
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Strong Convergence ◽  
Evolution Equations ◽  
Martensitic Transformations ◽  
Nonlinear Damping ◽  
Nonlinear Evolution Equations ◽  
Discrete Approximations ◽  
Partial Differential ◽  
Vibrating Membrane

Abstract Recent results on convergence of fully discrete approximations combining the Galerkin method with the explicit-implicit Euler scheme are extended to strong convergence under additional monotonicity assumptions. It is shown that these abstract results, formulated in the setting of evolution equations, apply, for example, to the partial differential equation for vibrating membrane with nonlinear damping and to another partial differential equation that is similar to one of the equations used to describe martensitic transformations in shape-memory alloys. Numerical experiments are performed for the vibrating membrane equation with nonlinear damping which support the convergence results.

Sign in / Sign up

Export Citation Format

Share Document