2,354 research outputs found
Analysis for time discrete approximations of blow-up solutions of semilinear parabolic equations
We prove a posteriori error estimates for time discrete approximations, for semilinear parabolic equations with solutions that might blow up in finite time. In particular we consider the backward Euler and the Crank–Nicolson methods. The main tools that are used in the analysis are the reconstruction technique and energy methods combined with appropriate fixed point arguments. The final estimates we derive are conditional and lead to error control near the blow up time
Time--space white noise eliminates global solutions in reaction diffusion equations
We prove that perturbing the reaction--diffusion equation (), with time--space white noise produces that solutions explodes
with probability one for every initial datum, opposite to the deterministic
model where a positive stationary solution exists.Comment: New results included. To be published in Physica
Structures and waves in a nonlinear heat-conducting medium
The paper is an overview of the main contributions of a Bulgarian team of
researchers to the problem of finding the possible structures and waves in the
open nonlinear heat conducting medium, described by a reaction-diffusion
equation. Being posed and actively worked out by the Russian school of A. A.
Samarskii and S.P. Kurdyumov since the seventies of the last century, this
problem still contains open and challenging questions.Comment: 23 pages, 13 figures, the final publication will appear in Springer
Proceedings in Mathematics and Statistics, Numerical Methods for PDEs:
Theory, Algorithms and their Application
An adaptive space-time Newton–Galerkin approach for semilinear singularly perturbed parabolic evolution equations
Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)In this article, we develop an adaptive procedure for the numerical solution of semilinear parabolic problems with possible singular perturbations. Our approach combines a linearization technique using Newton’s method with an adaptive discretization – which is based on a spatial finite element method and the backward Euler time-stepping scheme – of the resulting sequence of linear problems. Upon deriving a robust a posteriori error analysis, we design a fully adaptive Newton-Galerkin time-stepping algorithm. Numerical experiments underline the robustness and reliability of the proposed approach for various examples
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