Finite element approximation of a model reaction ? diffusion problem with a non-Lipschitz nonlinearity

1991 ◽  
Vol 59 (1) ◽  
pp. 217-242 ◽  
Author(s):  
John W. Barrett ◽  
Robert M. Shanahan
Keyword(s):  
Finite Element ◽  
Reaction Diffusion ◽  
Finite Element Approximation ◽  
Diffusion Problem ◽  
Model Reaction ◽  
Element Approximation ◽  
Lipschitz Nonlinearity

Numerical Analysis for a Nonlocal Parabolic Problem

2016 ◽  
Vol 6 (4) ◽  
pp. 434-447 ◽  
Author(s):  
M. Mbehou ◽  
R. Maritz ◽  
P.M.D. Tchepmo
Keyword(s):  
Finite Element ◽  
Parabolic Problem ◽  
Reaction Diffusion ◽  
Finite Element Approximation ◽  
Reaction Diffusion Equation ◽  
Element Approximation ◽  
Galerkin Finite Element ◽  
Fully Discrete ◽  
Nonlinear Reaction Diffusion Equation ◽  
Nonlinear Parabolic

AbstractThis article is devoted to the study of the finite element approximation for a nonlocal nonlinear parabolic problem. Using a linearised Crank-Nicolson Galerkin finite element method for a nonlinear reaction-diffusion equation, we establish the convergence and error bound for the fully discrete scheme. Moreover, important results on exponential decay and vanishing of the solutions in finite time are presented. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

scholarly journals Nonlinearly Preconditioned FETI Solver for Substructured Formulations of Nonlinear Problems

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3165
Author(s):  
Camille Negrello ◽  
Pierre Gosselet ◽  
Christian Rey
Keyword(s):  
Finite Element ◽  
Time Integration ◽  
Nonlinear Problems ◽  
Finite Element Approximation ◽  
Diffusion Problem ◽  
Implicit Time ◽  
Element Approximation ◽  
Dual Approach ◽  
Nonlinear Version ◽  
Exact Shape

We consider the finite element approximation of the solution to elliptic partial differential equations such as the ones encountered in (quasi)-static mechanics, in transient mechanics with implicit time integration, or in thermal diffusion. We propose a new nonlinear version of preconditioning, dedicated to nonlinear substructured and condensed formulations with dual approach, i.e., nonlinear analogues to the Finite Element Tearing and Interconnecting (FETI) solver. By increasing the importance of local nonlinear operations, this new technique reduces communications between processors throughout the parallel solving process. Moreover, the tangent systems produced at each step still have the exact shape of classically preconditioned linear FETI problems, which makes the tractability of the implementation barely modified. The efficiency of this new preconditioner is illustrated on two academic test cases, namely a water diffusion problem and a nonlinear thermal behavior.

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