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Parallel Schwarz Waveform Relaxation Algorithm for an N-dimensional semilinear heat equation

Minh-Binh Tran (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We present in this paper a proof of well-posedness and convergence for the parallel Schwarz Waveform Relaxation Algorithm adapted to an N-dimensional semilinear heat equation. Since the equation we study is an evolution one, each subproblem at each step has its own local existence time, we then determine a common existence time for every problem in any subdomain at any step. We also introduce a new technique: Exponential Decay Error Estimates, to prove the convergence of the Schwarz Methods, with...

Parallel strategies for solving the FETI coarse problem in the PERMON toolbox

Vašatová, Alena, Tomčala, Jiří, Sojka, Radim, Pecha, Marek, Kružík, Jakub, Horák, David, Hapla, Václav, Čermák, Martin (2017)

Programs and Algorithms of Numerical Mathematics

PERMON (Parallel, Efficient, Robust, Modular, Object-oriented, Numerical) is a newly emerging collection of software libraries, uniquely combining Quadratic Programming (QP) algorithms and Domain Decomposition Methods (DDM). Among the main applications are contact problems of mechanics. This paper gives an overview of PERMON and selected ingredients improving scalability, demonstrated by numerical experiments.

Particle-in-wavelets scheme for the 1D Vlasov-Poisson equations ⋆⋆⋆

Romain Nguyen van yen, Éric Sonnendrücker, Kai Schneider, Marie Farge (2011)

ESAIM: Proceedings

A new numerical scheme called particle-in-wavelets is proposed for the Vlasov-Poisson equations, and tested in the simplest case of one spatial dimension. The plasma distribution function is discretized using tracer particles, and the charge distribution is reconstructed using wavelet-based density estimation. The latter consists in projecting the Delta distributions corresponding to the particles onto a finite dimensional linear space spanned by...

Perona-Malik equation: properties of explicit finite volume scheme

Angela Handlovičová (2007)

Kybernetika

The Perona–Malik nonlinear parabolic problem, which is widely used in image processing, is investigated in this paper from the numerical point of view. An explicit finite volume numerical scheme for this problem is presented and consistency property is proved.

Phase field model for mode III crack growth in two dimensional elasticity

Takeshi Takaishi, Masato Kimura (2009)

Kybernetika

A phase field model for anti-plane shear crack growth in two dimensional isotropic elastic material is proposed. We introduce a phase field to represent the shape of the crack with a regularization parameter ϵ > 0 and we approximate the Francfort–Marigo type energy using the idea of Ambrosio and Tortorelli. The phase field model is derived as a gradient flow of this regularized energy. We show several numerical examples of the crack growth computed with an adaptive mesh finite element method.

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