Approximation of solution branches for semilinear bifurcation problems
Approximation of solution branches for semilinear bifurcation problems
This note deals with the approximation, by a finite element method with numerical integration, of solution curves of a semilinear problem. Because of both mixed boundary conditions and geometrical properties of the domain, some of the solutions do not belong to . So, classical results for convergence lead to poor estimates. We show how to improve such estimates with the use of weighted Sobolev spaces together with a mesh “ adapted” to the singularity. For the...
On the Caginalp system with dynamic boundary conditions and singular potentials
This article is devoted to the study of the Caginalp phase field system with dynamic boundary conditions and singular potentials. We first show that, for initial data in , the solutions are strictly separated from the singularities of the potential. This turns out to be our main argument in the proof of the existence and uniqueness of solutions. We then prove the existence of global attractors. In the last part of the article, we adapt well-known results concerning the Łojasiewicz inequality in...
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